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 [r]
(1)How to prove Pythagore theorem Proofs
This is a theorem that may have more known proofs than any other; the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs
[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
(2)[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
(3)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 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
[edit] Garfield's proof
(4)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
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
(5)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 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.[1]
[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
(6)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
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
(7)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, 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
A Pythagorean triple has positive numbers a, b, and c, such that a2 + b2 = c2 In other words, a
(8)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 The Pythagoreans proved that the square root of 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
[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
proof[1] r.[1]