PYTHAGORAS

Pythagoras
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Pythagoras
Pre-Socratic philosophy


Bust of Pythagoras of Samos in the Capitoline Museums, Rome

Name Pythagoras (Πυθαγόρας)
Birth c. 580 BC – 572 BC

Death c. 500 BC – 490 BC

School/tradition Pythagoreanism

Main interests Metaphysics, Music, Mathematics, Ethics, Politics

Notable ideas Musica universalis, Golden ratio, Pythagorean tuning, Pythagorean theorem

Influenced by Thales, Anaximander, Pherecydes

Influenced Philolaus, Alcmaeon, Parmenides, Plato, Euclid, Empedocles, Hippasus, Kepler

"Pythagoras of Samos" redirects here. For the Samian statuary of the same name, see Pythagoras (sculptor).
Pythagoras of Samos (Greek: Ὁ Πυθαγόρας ὁ Σάμιος, Pythagoras the Samian, or simply Ὁ Πυθαγόρας; born between 580 and 572 BC, died between 500 and 490 BC) was an Ionian Greek mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mystic and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy. Herodotus referred to him as "the most able philosopher among the Greeks". His name led him to be associated with Pythian Apollo; Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and of benefit to humankind.[1]
He is best known for the Pythagorean theorem, which bears his name. Known as "the father of numbers", Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. Because legend and obfuscation cloud his work even more than with the other pre-Socratics, one can say little with confidence about his life and teachings. We do know that Pythagoras and his students believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles. According to Iamblichus, Pythagoras once said that "number is the ruler of forms and ideas and the cause of gods and demons."
He was the first man to call himself a philosopher, or lover of wisdom,[2] and Pythagorean ideas exercised a marked influence on Plato. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors.
Contents
[hide]
• 1 Life
• 2 Pythagoreans
o 2.1 Musical theories and investigations
• 3 Influence
• 4 Religion and science
• 5 Literary works
• 6 Lore
• 7 Other accomplishments
• 8 Groups influenced by Pythagoras
o 8.1 Influence on Plato
o 8.2 Roman influence
o 8.3 Influence on esoteric groups
• 9 See also
• 10 Notes
• 11 References
o 11.1 Primary sources
o 11.2 Classical secondary sources
o 11.3 Modern secondary sources
• 12 External links

Life
Pythagoras was born on Samos, a Greek island in the eastern Aegean, off the coast of Asia Minor. He was born to Pythais (his mother, a native of Samos) and Mnesarchus (his father, a Phoenician merchant from Tyre). As a young man, he left his native city for Croton, Calabria, in Southern Italy, to escape the tyrannical government of Polycrates. According to Iamblichus, Thales, impressed with his abilities, advised Pythagoras to head to Memphis in Egypt and study with the priests there who were renowned for their wisdom. He was also discipled in the temples of Tyre and Byblos in Phoenicia. It may have been in Egypt where he learned some geometric principles which eventually inspired his formulation of the theorem that is now called by his name. This possible inspiration is presented as an extraordinaire problem in the Berlin Papyrus. Upon his migration from Samos to Croton, Calabria, Italy, Pythagoras established a secret religious society very similar to (and possibly influenced by) the earlier Orphic cult.


Bust of Pythagoras, Vatican
Pythagoras undertook a reform of the cultural life of Croton, urging the citizens to follow virtue and form an elite circle of followers around himself called Pythagoreans. Very strict rules of conduct governed this cultural center. He opened his school to both male and female students uniformly. Those who joined the inner circle of Pythagoras's society called themselves the Mathematikoi. They lived at the school, owned no personal possessions and were required to assume a mainly vegetarian diet (meat that could be sacrificed was allowed to be eaten). Other students who lived in neighboring areas were also permitted to attend Pythagoras's school. Known as Akousmatikoi, these students were permitted to eat meat and own personal belongings. Richard Blackmore, in his book The Lay Monastery (1714), saw in the religious observances of the Pythagoreans, "the first instance recorded in history of a monastic life."
According to Iamblichus, the Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study. Music featured as an essential organizing factor of this life: the disciples would sing hymns to Apollo together regularly; they used the lyre to cure illness of the soul or body; poetry recitations occurred before and after sleep to aid the memory.
Flavius Josephus, in his polemical Against Apion, in defence of Judaism against Greek philosophy, mentions that according to Hermippus of Smyrna, Pythagoras was familiar with Jewish beliefs, incorporating some of them in his own philosophy.
Towards the end of his life he fled to Metapontum because of a plot against him and his followers by a noble of Croton named Cylon. He died in Metapontum around 90 years old from unknown causes.
Pythagoreans
Main article: Pythagoreans


Pythagoras, the man in the center with the book, teaching music, in The School of Athens by Raphael
The organization was in some ways a school, in some ways a brotherhood, and in some ways a monastery. It was based upon the religious teachings of Pythagoras and was very secretive. At first, the school was highly concerned with the morality of society. Members were required to live ethically, love one another, share political beliefs, practice pacifism, and devote themselves to the mathematics of nature.
Pythagoras's followers were commonly called "Pythagoreans". They are generally accepted as philosophical mathematicians who had an influence on the beginning of axiomatic geometry, which after two hundred years of development was written down by Euclid in The Elements.
The Pythagoreans observed a rule of silence called echemythia, the breaking of which was punishable by death. This was because the Pythagoreans believed that a man's words were usually careless and misrepresented him and that when someone was "in doubt as to what he should say, he should always remain silent". Another rule that they had was to help a man "in raising a burden, but do not assist him in laying it down, for it is a great sin to encourage indolence", and they said "departing from your house, turn not back, for the furies will be your attendants"; this axiom reminded them that it was better to learn none of the truth about mathematics, God, and the universe at all than to learn a little without learning all. (The Secret Teachings of All Ages by Manly P. Hall).
In his biography of Pythagoras (written seven centuries after Pythagoras's time), Porphyry stated that this silence was "of no ordinary kind." The Pythagoreans were divided into an inner circle called the mathematikoi ("mathematicians") and an outer circle called the akousmatikoi ("listeners"). Porphyry wrote "the mathematikoi learned the more detailed and exactly elaborated version of this knowledge, the akousmatikoi (were) those who had heard only the summary headings of his (Pythagoras's) writings, without the more exact exposition." According to Iamblichus, the akousmatikoi were the exoteric disciples who listened to lectures that Pythagoras gave out loud from behind a veil.
The akousmatikoi were not allowed to see Pythagoras and they were not taught the inner secrets of the cult. Instead they were taught laws of behavior and morality in the form of cryptic, brief sayings that had hidden meanings. The akousmatikoi recognized the mathematikoi as real Pythagoreans, but not vice versa. After the murder of a number of the mathematikoi by the cohorts of Cylon, a resentful disciple, the two groups split from each other entirely, with Pythagoras's wife Theano and their two daughters leading the mathematikoi.
Theano, daughter of the Orphic initiate Brontinus, was a mathematician in her own right. She is credited with having written treatises on mathematics, physics, medicine, and child psychology, although nothing of her writing survives. Her most important work is said to have been a treatise on the principle of the golden mean. In a time when women were usually considered property and relegated to the role of housekeeper or spouse, Pythagoras allowed women to function on equal terms in his society.
The Pythagorean society is associated with prohibitions such as not to step over a crossbar, and not to eat beans. These rules seem like primitive superstition, similar to "walking under a ladder brings bad luck". The abusive epithet mystikos logos ("mystical speech") was hurled at Pythagoras even in ancient times to discredit him. The prohibition on beans could be linked to favism, which is relatively widespread around the Mediterranean.
The key here is that akousmata means "rules", so that the superstitious taboos primarily applied to the akousmatikoi, and many of the rules were probably invented after Pythagoras's death and independent from the mathematikoi (arguably the real preservers of the Pythagorean tradition). The mathematikoi placed greater emphasis on inner understanding than did the akousmatikoi, even to the extent of dispensing with certain rules and ritual practices. For the mathematikoi, being a Pythagorean was a question of innate quality and inner understanding.
There was also another way of dealing with the akousmata — by allegorizing them. We have a few examples of this, one being Aristotle's explanations of them: "'step not over a balance', i.e. be not covetous; 'poke not the fire with a sword', i.e. do not vex with sharp words a man swollen with anger, 'eat not heart', i.e. do not vex yourself with grief," etc. We have evidence for Pythagoreans allegorizing in this way at least as far back as the early fifth century BC. This suggests that the strange sayings were riddles for the initiated.
The Pythagoreans are known for their theory of the transmigration of souls, and also for their theory that numbers constitute the true nature of things. They performed purification rites and followed and developed various rules of living which they believed would enable their soul to achieve a higher rank among the gods.
Much of their mysticism concerning the soul seem inseparable from the Orphic tradition. The Orphics advocated various purificatory rites and practices as well as incubatory rites of descent into the underworld. Pythagoras is also closely linked with Pherecydes of Syros, the man ancient commentators tend to credit as the first Greek to teach a transmigration of souls. Ancient commentators agree that Pherekydes was Pythagoras's most intimate teacher. Pherekydes expounded his teaching on the soul in terms of a pentemychos ("five-nooks", or "five hidden cavities") — the most likely origin of the Pythagorean use of the pentagram, used by them as a symbol of recognition among members and as a symbol of inner health (ugieia).
Musical theories and investigations
Pythagoras was very interested in music, and so were his followers. The Pythagoreans were musicians as well as mathematicians. Pythagoras wanted to improve the music of his day, which he believed was not harmonious enough and was too hectic.
According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when one day he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how this had happened by looking at their tools, he discovered that it was because the anvils were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on." (See Pythagorean tuning.)
The Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Pythagoras believed in something called the harmony of the spheres. He believed that the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.[3]
Academic Genealogy
Notable teachers Notable students
Anaximander
Pherecydes of Syros
Hermodamas of Samos
Thales
Ameinias
Bathyllus
Brontinus
Calliphon
Cercops
Echecrates
Empedocles
Eurytus
Hippasus
Leon
Lysis of Tarentum
Milon, whose house was used as a Pythagorean meeting place
Parmeniscus
Petron
Philolaus of Croton
Theano, Pythagoras' Wife/Daughter of Milon
Xenophilus of Chaldice
Zalmoxis,

Influence


The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Pythagoras is commonly given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle), c, is equal to the sum of the squares of the other two sides, b and a—that is, a² + b² = c².
While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are thought to have constructed the first proof. Because of the secretive nature of his school and the custom of its students to attribute everything to their teacher, there is no evidence that Pythagoras himself worked on or proved this theorem. For that matter, there is no evidence that he worked on any mathematical or meta-mathematical problems. Some attribute it as a carefully constructed myth by followers of Plato over two centuries after the death of Pythagoras, mainly to bolster the case for Platonic meta-physics, which resonate well with the ideas they attributed to Pythagoras. This attribution has stuck, down the centuries up to modern times. [4] The earliest known mention of Pythagoras's name in connection with the theorem occurred five centuries after his death, in the writings of Cicero and Plutarch.
Today, Pythagoras is revered as a prophet by the Ahl al-Tawhid or Druze faith along with his fellow Greek, Plato. But Pythagoras also had his critics, such as Heraclitus who said that "much learning does not teach wisdom; otherwise it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus".[5]
Religion and science
Pythagoras’ religious and scientific views were, in his opinion, inseparably interconnected. However, they are looked at separately in the 21st century. Religiously, Pythagoras was a believer of metempsychosis. He believed in transmigration, or the reincarnation of the soul again and again into the bodies of humans, animals, or vegetables until it became moral. His ideas of reincarnation were influenced by ancient Greek religion. He was one of the first to propose that the thought processes and the soul were located in the brain and not the heart. He himself claimed to have lived four lives that he could remember in detail, and heard the cry of his dead friend in the bark of a dog.
One of Pythagoras' beliefs was that the essence of being is number. Thus, being relies on stability of all things that create the universe. Things like health relied on a stable proportion of elements; too much or too little of one thing causes an imbalance that makes a being unhealthy. Pythagoras viewed thinking as the calculating with the idea numbers. When combined with the Folk theories, the philosophy evolves into a belief that Knowledge of the essence of being can be found in the form of numbers. If this is taken a step further, one can say that because mathematics is an unseen essence, the essence of being is an unseen characteristic that can be encountered by the study of mathematics.
Literary works
No texts by Pythagoras survive, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching. Pythagoras appears as a character in the last book of Ovid's Metamorphoses, where Ovid has him expound upon his philosophical viewpoints. Pythagoras has been quoted as saying, "No man is free who cannot command himself."
Lore
There is another side to Pythagoras, as he became the subject of elaborate legends surrounding his historic persona. Aristotle described Pythagoras as wonder-worker and somewhat of a supernatural figure, attributing to him such aspects as a golden thigh, which was a sign of divinity. According to Aristotle and others' accounts, some ancients believed that he had the ability to travel through space and time, and to communicate with animals and plants.[6] An extract from Brewer's Dictionary of Phrase and Fable's entry entitled "Golden Thigh":
Pythagoras is said to have had a golden thigh, which he showed to Abaris, the Hyperborean priest, and exhibited in the Olympic games.[7]
Another legend, also taken from Brewer's Dictionary, describes his writing on the moon:
Pythagoras asserted he could write on the moon. His plan of operation was to write on a looking-glass in blood, and place it opposite the moon, when the inscription would appear photographed or reflected on the moon's disc.[8]
Other accomplishments
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The number is irrational.
One of Pythagoras's major accomplishments was the discovery that music was based on proportional intervals of the numbers one through four. He believed that the number system, and therefore the universe system, was based on the sum of these numbers: ten. Pythagoreans swore by the Tetrachtys of the Decad, or ten, rather than by the gods. Odd numbers were masculine and even were feminine. He discovered the theory of mathematical proportions, constructed from three to five geometrical solids. One of his order, Hippasos, also discovered irrational numbers, but the idea was unthinkable to Pythagoras, and according to one version this member was executed. Pythagoras (or the Pythagoreans) also discovered square numbers. They found that if one took, for example, four small stones and arranged them into a square, each side of the square was not only equivalent to the other, but that when the two sides were multiplied together, they equaled the sum total of stones in the square arrangement, hence the name "Square Root"[9]. He was one of the first to think that the earth was round, that all planets have an axis, and that all the planets travel around one central point. He originally identified that point as Earth, but later renounced it for the idea that the planets revolve around a central “fire” that he never identified as the sun. He also believed that the moon was another planet that he called a “counter-Earth” – furthering his belief in the Limited-Unlimited.
Groups influenced by Pythagoras
Influence on Plato
Pythagoras or in a broader sense, the Pythagoreans, allegedly exercised an important influence on the work of Plato. According to R. M. Hare, his influence consists of three points: a) the platonic Republic might be related to the idea of "a tightly organized community of like-minded thinkers", like the one established by Pythagoras in Croton. b) there is evidence that Plato possibly took from Pythagoras the idea that mathematics and, generally speaking, abstract thinking is a secure basis for philosophical thinking as well as "for substantial theses in science and morals". c) Plato and Pythagoras shared a "mystical approach to the soul and its place in the material world". It is probable that both have been influenced by Orphism.[10]
Plato's harmonics were clearly influenced by the work of Archytas, a genuine Pythagorean of the third generation, who made important contributions to geometry, reflected in Book VIII of Euclid's Elements.
Roman influence
In the legends of ancient Rome, Numa Pompilius, the second King of Rome, is said to have studied under Pythagoras. This is unlikely, since the commonly accepted dates for the two lives do not overlap.
Influence on esoteric groups
Pythagoras started a secret society called the Pythagorean brotherhood devoted to the study of mathematics. This had a great effect on future esoteric traditions, such as Rosicrucianism and Freemasonry, both of which were occult groups dedicated to the study of mathematics and both of which claimed to have evolved out of the Pythagorean brotherhood. The mystical and occult qualities of Pythagorean mathematics are discussed in a chapter of Manly P. Hall's The Secret Teachings of All Ages entitled "Pythagorean Mathematics".
Pythagorean theory was tremendously influential on later numerology, which was extremely popular throughout the Middle East in the ancient world. The 8th-century Muslim alchemist Jabir ibn Hayyan grounded his work in an elaborate numerology greatly influenced by Pythagorean theory.
Pythagorean theorem
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The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
In mathematics, the Pythagorean theorem (American English) or Pythagoras' theorem (British English) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof,[1] although knowledge of the theorem almost certainly predates him. The theorem is as follows:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
This is usually summarized as follows:
The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.
If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation:

or, solved for c:

If c is already given, and the length of one of the legs must be found, the following equations can be used (The following equations are simply the converse of the original equation):

or

This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle it reduces to the Pythagorean theorem.


Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Trigonometry

History
Usage
Functions
Inverse functions
Further reading

Reference
List of identities
Exact constants
Generating trigonometric tables
CORDIC

Euclidean theory

Law of sines
Law of cosines
Law of tangents
Pythagorean theorem
Calculus

The Trigonometric integral
Trigonometric substitution
Integrals of functions
Integrals of inverses

Contents
[hide]
• 1 History
• 2 Proofs
o 2.1 Proof using similar triangles
o 2.2 Euclid's proof
o 2.3 Garfield's proof
o 2.4 Similarity proof
o 2.5 Proof by rearrangement
o 2.6 Algebraic proof
o 2.7 Proof by differential equations
• 3 Converse
• 4 Consequences and uses of the theorem
o 4.1 Pythagorean triples
o 4.2 List of primitive Pythagorean triples up to 100
o 4.3 The existence of irrational numbers
o 4.4 Distance in Cartesian coordinates
• 5 Generalizations
o 5.1 The Pythagorean theorem in non-Euclidean geometry
• 6 Cultural references to the Pythagorean theorem
• 7 See also
• 8 Notes
• 9 References
• 10 External links

[edit] History
This section needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (April 2008)
The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, knowledge of the relationship between adjacent angles, and proofs of the theorem.
Megalithic monuments from circa 2500 BC in Egypt, and in Northern Europe, incorporate right triangles with integer sides.[2] Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically.[3]
Written between 2000 and 1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem whose solution is a Pythagorean triple.
During the reign of Hammurabi the Great, the Mesopotamian tablet Plimpton 322, written between 1790 and 1750 BC, contains many entries closely related to Pythagorean triples.
The Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th century BC and the 2nd century BC, in India, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle.
The Apastamba Sulba Sutra (circa 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert Bŭrk, this is the original proof of the theorem; he further theorizes that Pythagoras visited Arakonam, India, and copied it.
Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proklos's commentary on Euclid. Proklos, however, wrote between 410 and 485 AD. According to Sir Thomas L. Heath, there is no attribution of the theorem to Pythagoras for five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.[4]
Around 400 BC, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Circa 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.
Written sometime between 500 BC and 200 AD, the Chinese text Chou Pei Suan Ching (周髀算经), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives a visual proof of the Pythagorean theorem — in China it is called the "Gougu Theorem" (勾股定理) — for the (3, 4, 5) triangle. During the Han Dynasty, from 202 BC to 220 AD, Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles.[5]
The first recorded use is in China, known as the "Gougu theorem" (勾股定理) and in India known as the Bhaskara Theorem.
There is much debate on whether the Pythagorean theorem was discovered once or many times. Boyer (1991) thinks the elements found in the Shulba Sutras may be of Mesopotamian derivation.[6]
[edit] Proofs
This is a theorem that may have more known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs.
Some arguments based on trigonometric identities (such as Taylor series for sine and cosine) have been proposed as proofs for the theorem. However, since all the fundamental trigonometric identities are proved using the Pythagorean theorem, there cannot be any trigonometric proof. (See also begging the question.)
[edit] Proof using similar triangles


Proof using similar triangles.
Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As

so

These can be written as

Summing these two equalities, we obtain

In other words, the Pythagorean theorem:

[edit] Euclid's proof


Proof in Euclid's Elements
In Euclid's Elements, Proposition 47 of Book 1, the Pythagorean theorem is proved by an argument along the following lines. Let A, B, C be the vertices of a right triangle, with a right angle at A. Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.
For the formal proof, we require four elementary lemmata:
1. If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent. (Side - Angle - Side Theorem)
2. The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
3. The area of any square is equal to the product of two of its sides.
4. The area of any rectangle is equal to the product of two adjacent sides (follows from Lemma 3).
The intuitive idea behind this proof, which can make it easier to follow, is that the top squares are morphed into parallelograms with the same size, then turned and morphed into the left and right rectangles in the lower square, again at constant area.
The proof is as follows:
1. Let ACB be a right-angled triangle with right angle CAB.
2. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order.
3. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
4. Join CF and AD, to form the triangles BCF and BDA.


Illustration including the new lines
1. Angles CAB and BAG are both right angles; therefore C, A, and G are collinear. Similarly for B, A, and H.
2. Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC, since both are the sum of a right angle and angle ABC.
3. Since AB and BD are equal to FB and BC, respectively, triangle ABD must be equal to triangle FBC.
4. Since A is collinear with K and L, rectangle BDLK must be twice in area to triangle ABD.
5. Since C is collinear with A and G, square BAGF must be twice in area to triangle FBC.
6. Therefore rectangle BDLK must have the same area as square BAGF = AB2.
7. Similarly, it can be shown that rectangle CKLE must have the same area as square ACIH = AC2.
8. Adding these two results, AB2 + AC2 = BD × BK + KL × KC
9. Since BD = KL, BD* BK + KL × KC = BD(BK + KC) = BD × BC
10. Therefore AB2 + AC2 = BC2, since CBDE is a square.
This proof appears in Euclid's Elements as that of Proposition 1.47.[7]
[edit] Garfield's proof
James A. Garfield (later President of the United States) is credited with a novel algebraic proof[1] using a trapezoid containing two examples of the triangle, the figure comprising one-half of the figure using four triangles enclosing a square shown below.


Proof using area subtraction.
[edit] Similarity proof
From the same diagram as that in Euclid's proof above, we can see three similar figures, each being "a square with a triangle on top". Since the large triangle is made of the two smaller triangles, its area is the sum of areas of the two smaller ones. By similarity, the three squares are in the same proportions relative to each other as the three triangles, and so likewise the area of the larger square is the sum of the areas of the two smaller squares.
[edit] Proof by rearrangement


Proof of Pythagorean theorem by rearrangement of 4 identical right triangles. Since the total area and the areas of the triangles are all constant, the total black area is constant. But this can be divided into squares delineated by the triangle sides a, b, c, demonstrating that a2 + b2 = c2 .
A proof by rearrangement is given by the illustration and the animation. In the illustration, the area of each large square is (a + b)2. In both, the area of four identical triangles is removed. The remaining areas, a2 + b2 and c2, are equal. Q.E.D.


Animation showing another proof by rearrangement.


Proof using rearrangement.


A square created by aligning four right angle triangles and a large square.
This proof is indeed very simple, but it is not elementary, in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometry. In particular, while it is quite easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself (see Lebesgue measure and Banach-Tarski paradox). Actually, this difficulty affects all simple Euclidean proofs involving area; for instance, deriving the area of a right triangle involves the assumption that it is half the area of a rectangle with the same height and base. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see above).
A third graphic illustration of the Pythagorean theorem (in yellow and blue to the right) fits parts of the sides' squares into the hypotenuse's square. A related proof would show that the repositioned parts are identical with the originals and, since the sum of equals are equal, that the corresponding areas are equal. To show that a square is the result one must show that the length of the new sides equals c. Note that for this proof to work, one must provide a way to handle cutting the small square in more and more slices as the corresponding side gets smaller and smaller.[8]
[edit] Algebraic proof
An algebraic variant of this proof is provided by the following reasoning. Looking at the illustration which is a large square with identical right triangles in its corners, the area of each of these four triangles is given by an angle corresponding with the side of length C.

The A-side angle and B-side angle of each of these triangles are complementary angles, so each of the angles of the blue area in the middle is a right angle, making this area a square with side length C. The area of this square is C2. Thus the area of everything together is given by:

However, as the large square has sides of length A + B, we can also calculate its area as (A + B)2, which expands to A2 + 2AB + B2.

(Distribution of the 4)
(Subtraction of 2AB)
[edit] Proof by differential equations
One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse in the following diagram and employing a little calculus.[9]


Proof using differential equations.
As a result of a change in side a,

by similar triangles and for differential changes. So

upon separation of variables.
which results from adding a second term for changes in side b.
Integrating gives

When a = 0 then c = b, so the "constant" is b2. So

As can be seen, the squares are due to the particular proportion between the changes and the sides while the sum is a result of the independent contributions of the changes in the sides which is not evident from the geometric proofs. From the proportion given it can be shown that the changes in the sides are inversely proportional to the sides. The differential equation suggests that the theorem is due to relative changes and its derivation is nearly equivalent to computing a line integral.
These quantities da and dc are respectively infinitely small changes in a and c. But we use instead real numbers Δa and Δc, then the limit of their ratio as their sizes approach zero is da/dc, the derivative, and also approaches c/a, the ratio of lengths of sides of triangles, and the differential equation results.
[edit] Converse
The converse of the theorem is also true:
For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.
This converse also appears in Euclid's Elements. It can be proven using the law of cosines (see below under Generalizations), or by the following proof:
Let ABC be a triangle with side lengths a, b, and c, with a2 + b2 = c2. We need to prove that the angle between the a and b sides is a right angle. We construct another triangle with a right angle between sides of lengths a and b. By the Pythagorean theorem, it follows that the hypotenuse of this triangle also has length c. Since both triangles have the same side lengths a, b and c, they are congruent, and so they must have the same angles. Therefore, the angle between the side of lengths a and b in our original triangle is a right angle.
A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Where c is chosen to be the longest of the three sides:
• If a2 + b2 = c2, then the triangle is right.
• If a2 + b2 > c2, then the triangle is acute.
• If a2 + b2 < c2, then the triangle is obtuse.
[edit] Consequences and uses of the theorem
[edit] Pythagorean triples
Main article: Pythagorean triple
A Pythagorean triple has 3 positive numbers a, b, and c, such that a2 + b2 = c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Evidence from megalithic monuments on the Northern Europe shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).
[edit] List of primitive Pythagorean triples up to 100
(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)
[edit] The existence of irrational numbers
One of the consequences of the Pythagorean theorem is that irrational numbers, such as the square root of 2, can be constructed. A right triangle with legs both equal to one unit has hypotenuse length square root of 2. The Pythagoreans proved that the square root of 2 is irrational, and this proof has come down to us even though it flew in the face of their cherished belief that everything was rational. According to the legend, Hippasus, who first proved the irrationality of the square root of two, was drowned at sea as a consequence.[10]
[edit] Distance in Cartesian coordinates
The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If (x0, y0) and (x1, y1) are points in the plane, then the distance between them, also called the Euclidean distance, is given by

More generally, in Euclidean n-space, the Euclidean distance between two points, and , is defined, using the Pythagorean theorem, as:

[edit] Generalizations
The Pythagorean theorem was generalized by Euclid in his Elements:
If one erects similar figures (see Euclidean geometry) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.
The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:

where θ is the angle between sides a and b.
When θ is 90 degrees, then cos(θ) = 0, so the formula reduces to the usual Pythagorean theorem.
Given two vectors v and w in a complex inner product space, the Pythagorean theorem takes the following form:

In particular, ||v + w||2 = ||v||2 + ||w||2 if v and w are orthogonal, although the converse is not necessarily true.
Using mathematical induction, the previous result can be extended to any finite number of pairwise orthogonal vectors. Let v1, v2,…, vn be vectors in an inner product space such that = 0 for 1 ≤ i < j ≤ n. Then

The generalization of this result to infinite-dimensional real inner product spaces is known as Parseval's identity.
When the theorem above about vectors is rewritten in terms of solid geometry, it becomes the following theorem. If lines AB and BC form a right angle at B, and lines BC and CD form a right angle at C, and if CD is perpendicular to the plane containing lines AB and BC, then the sum of the squares of the lengths of AB, BC, and CD is equal to the square of AD. The proof is trivial.
Another generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (a corner like a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces.
There are also analogs of these theorems in dimensions four and higher.
In a triangle with three acute angles, α + β > γ holds. Therefore, a2 + b2 > c2 holds.
In a triangle with an obtuse angle, α + β < γ holds. Therefore, a2 + b2 < c2 holds.
Edsger Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:
sgn(α + β − γ) = sgn(a2 + b2 − c2)
where α is the angle opposite to side a, β is the angle opposite to side b and γ is the angle opposite to side c.[11]
[edit] The Pythagorean theorem in non-Euclidean geometry
The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. (It has been shown in fact to be equivalent to Euclid's Parallel (Fifth) Postulate.) For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to ; this violates the Euclidean Pythagorean theorem because .
This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean theorem. There are two cases to consider — spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case, the result follows from the appropriate law of cosines:
For any right triangle on a sphere of radius R, the Pythagorean theorem takes the form

By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form.
For any triangle in the hyperbolic plane (with Gaussian curvature −1), the Pythagorean theorem takes the form

where cosh is the hyperbolic cosine.
By using the Maclaurin series for this function, it can be shown that as a hyperbolic triangle becomes very small (i.e., as a, b, and c all approach zero), the hyperbolic form of the Pythagorean theorem approaches the Euclidean form.
In hyperbolic geometry, for a right triangle one can also write,

where is the angle of parallelism of the line segment AB that where μ is the multiplicative distance function (see Hilbert's arithmetic of ends).
In hyperbolic trigonometry, the sine of the angle of parallelism satisfies

Thus, the equation takes the form

where a, b, and c are multiplicative distances of the sides of the right triangle (Hartshorne, 2000).
Special right triangles
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Two types of special right triangles appear commonly in geometry, the "angle based" and the "side based" (or Pythagorean) triangles. The former are characterised by integer ratios between the triangle angles, and the latter by integer ratios between the sides. Knowing the ratios of the sides of these special right triangles allows one to quickly calculate various lengths in geometric problems.
Contents
[hide]
• 1 Angle-based
o 1.1 45-45-90 triangle
o 1.2 30-60-90 triangle
• 2 Side-based
o 2.1 Common Pythagorean triples
o 2.2 Fibonacci triangles
o 2.3 Almost-isosceles Pythagorean triples
• 3 See also
• 4 External links
• 5 References

[edit] Angle-based
"Angle-based" special right triangles are specified by the integer ratio of the angles of which the triangle is composed. The integer ratio of the angles of these triangles are such that the larger (right) angle equals the sum of the smaller angles: . The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This form is most interesting in that it may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, & 60°.
[edit] 45-45-90 triangle


The side lengths of a 45-45-90 triangle
Constructing the diagonal of a square results in a triangle whose three angles are in the ratio . With the three angles adding up to 180°, the angles respectively measure 45°, 45°, and 90°. The sides are in the ratio

A simple proof. Say you have such a triangle with legs a and b and hypotenuse c. Suppose that a = 1. Since two angles measure 45°, this is an isosceles triangle and we have b = 1. The fact that follows immediately from the Pythagorean theorem.
[edit] 30-60-90 triangle


The side lengths of a 30-60-90 triangle
This is a triangle whose three angles are in the ratio , and respectively measure 30°, 60°, and 90°. The sides are in the ratio

The proof of this fact is clear using trigonometry. Although the geometric proof is less apparent, it is equally trivial:
Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30-60-90 triangle with hypotenuse of length 2, and base BD of length 1.
The fact that the remaining leg AD has length follows immediately from the Pythagorean theorem.


[edit] Side-based
All of the special side based right triangles possess angles which are not necessarily rational numbers, but whose sides are always of integer length and form a Pythagorean triple. They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship.
[edit] Common Pythagorean triples
There are several Pythagorean triples which are very well known, including:


(a multiple of the 3:4:5 triple)



The smallest of these (and its multiples, 6:8:10, 9:12:15, ...) is the only right triangle with edges in arithmetic progression. Triangles based on Pythagorean triplets are Heronian and therefore have integer area.
[edit] Fibonacci triangles
Starting with 5, every other Fibonacci number {0,1,1,2,3,5,8,13,21,34,55,89,...} is the length of the hypotenuse of a right triangle with integral sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.
The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely and approaches a limiting triangle with edge ratios:
.
This right triangle is sometimes referred to as a dom, a name suggested by Andrew Clarke to stress that this is the triangle obtained from dissecting a domino along a diagonal.
[edit] Almost-isosceles Pythagorean triples
Isosceles right-angled triangles can not have integral sides. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one.[1] Such almost-isosceles right-angled triangles can be obtained recursively using Pell's equation:
a0 = 1, b0 = 2
an = 2bn-1 + an-1
bn = 2an + bn-1
an is length of hypotenuse, n=1, 2, 3, .... The smallest Pythagorean triples resulting are:




Triangle
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Triangle (disambiguation).
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.
In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).


A triangle.
Contents
[hide]
• 1 Types of triangles
• 2 Basic facts
• 3 Points, lines and circles associated with a triangle
• 4 Computing the area of a triangle
o 4.1 Using vectors
o 4.2 Using trigonometry
o 4.3 Using coordinates
o 4.4 Using Heron's formula
• 5 Non-planar triangles
• 6 See also
• 7 References
• 8 External links

[edit] Types of triangles
Triangles can be classified according to the relative lengths of their sides:
• In an equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon.[1]
• In an isosceles triangle, two sides are of equal length (originally and conventionally limited to exactly two).[2] An isosceles triangle also has two equal angles: the angles opposite the two equal sides.
• In a scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.[3]

Equilateral Isosceles Scalene
Triangles can also be classified according to their internal angles, described below using degrees of arc:
• A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.
• An oblique triangle has no internal angle equal to 90°.
• An obtuse triangle is an oblique triangle with one internal angle larger than 90° (an obtuse angle).
• An acute triangle is an oblique triangle with internal angles all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.


Right Obtuse Acute

Oblique
[edit] Basic facts
Elementary facts about triangles were presented by Euclid in books 1-4 of his Elements around 300 BCE. A triangle is a polygon and a 2-simplex (see polytope). All triangles are two-dimensional.
The angles of a triangle in Euclidean space always add up to 180 degrees. An exterior angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to. Like all convex polygons, the exterior angles of a triangle add up to 360 degrees.
The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the triangle inequality. (In the special case of equality, two of the angles have collapsed to size zero, and the triangle has degenerated to a line segment.)
Two triangles are said to be similar if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are proportional. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.
A few basic postulates and theorems about similar triangles:
• Two triangles are similar if at least two corresponding angles are equal.
• If two corresponding sides of two triangles are in proportion, and their included angles are equal, the triangles are similar.
• If three sides of two triangles are in proportion, the triangles are similar.
For two triangles to be congruent, each of their corresponding angles and sides must be equal (6 total). A few basic postulates and theorems about congruent triangles:
• SAS Postulate: If two sides and the included angles of two triangles are correspondingly equal, the two triangles are congruent.
• SSS Postulate: If every side of two triangles are correspondingly equal, the triangles are congruent.
• ASA Postulate: If two angles and the included sides of two triangles are correspondingly equal, the two triangles are congruent.
• AAS Theorem: If two angles and any side of two triangles are correspondingly equal, the two triangles are congruent.
• Hypotenuse-Leg Theorem: If the hypotenuses and one leg of two right triangles are correspondingly equal, the triangles are congruent.
• Hypothenuse-Angle Theorem: If the hypothenuse and an acute angle of one right triangle are congruent to a hypothenuse and an acute angle of another right triangle, then the triangles are congruent
Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.
In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known.


The Pythagorean theorem
A central theorem is the Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that

The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle.
Some other facts about right triangles:
• The acute angles of a right triangle are complementary.
• If the legs of a right triangle are equal, then the angles opposite the legs are equal, acute and complementary, and thus are both 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times the square root of two.
• In a 30-60 right triangle, in which the acute angles measure 30 and 60 degrees, the hypotenuse is twice the length of the shorter side.
• In all right triangles, the median on the hypotenuse is the half of the hypotenuse.
For all triangles, angles and sides are related by the law of cosines and law of sines.
[edit] Points, lines and circles associated with a triangle
There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.

The circumcenter is the center of a circle passing through the three vertices of the triangle.
A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle can be found from the law of sines stated above.
Thales' theorem implies that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.

The intersection of the altitudes is the orthocenter.
An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. The three vertices together with the orthocenter are said to form an orthocentric system.

The intersection of the angle bisectors finds the center of the incircle.
An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.


The intersection of the medians is the centroid.
A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid. The centroid of a stiff triangular object (cut out of a thin sheet of uniform density) is also its center of gravity: the object can be balanced it on its centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side.

Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle.
The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.


Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).
The centroid (yellow), orthocenter (blue), circumcenter (green) and barycenter of the nine-point circle (red point) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
The center of the incircle is not in general located on Euler's line.
If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.

[edit] Computing the area of a triangle
Calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known, and simplest formula is

where S is area, b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term 'base' denotes any side, and 'height' denotes the length of a perpendicular from the point opposite the side onto the side itself.
Although simple, this formula is only useful if the height can be readily found. For example, the surveyor of a triangular field measures the length of each side, and can find the area from his results without having to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.[4]
[edit] Using vectors
The area of a parallelogram can be calculated using vectors. Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then |AB × AC|, which is the magnitude of the cross product of vectors AB and AC. |AB × AC| is equal to |h × AC|, where h represents the altitude h as a vector.
The area of triangle ABC is half of this, or S = ½|AB × AC|.
The area of triangle ABC can also be expressed in terms of dot products as follows:


Applying trigonometry to find the altitude h.
[edit] Using trigonometry
The height of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as:

Furthermore, since sin α = sin (π - α) = sin (β + γ), and similarly for the other two angles:

[edit] Using coordinates
If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xB, yB) and C = (xC, yC), then the area S can be computed as ½ times the absolute value of the determinant

For three general vertices, the equation is:

In three dimensions, the area of a general triangle {A = (xA, yA, zA), B = (xB, yB, zB) and C = (xC, yC, zC)} is the Pythagorean sum of the areas of the respective projections on the three principal planes (i.e. x = 0, y = 0 and z = 0):

[edit] Using Heron's formula
The shape of the triangle is determined by the lengths of the sides alone. Therefore the area S also can be derived from the lengths of the sides. By Heron's formula:

where s = ½ (a + b + c) is the semiperimeter, or half of the triangle's perimeter.
An equivalent way of writing Heron's formula is

[edit] Non-planar triangles
A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.
While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that the sum of its angles were greater than 180°.