Now, why would anyone devote virtual space to the Pythagorean Theorem? Because it's the most-proved theorem in history? Because it's one of the, if not the most important one in math? Nah... Mostly because we're bored over here at Kalish Inc.! And I actually think it's kinda cool how many ways you can prove it (there are > 200 known). This page shows a couple of them, including a couple of neat facts about it.

The Pythagorean Theorem states that for a right triangle with legs a and b with hypotenuse c, that
a² + b² = c²

One of the first proofs of the theorem, and perhaps the most famous, was Euclid's; which, ironically, is much less intuitive than more recent proofs.

- -- --- < Proof #1 > --- -- -

The setup:

right triangle ABC; CL -->>-- BE.

In the diagrams below I've demonstrated the proof:

• A shows two congruent triangles, AFB and ACD.

• B and C show each triangle equal in area to two other significant areas -- half of rectangle AMLD and square ACGF respectively. This is because the triangle shares a base with the quadrilateral, and has the same height.

• D: Doubles of equal are equal -- the square and rectangle are equal in area.

• P --> S A very similar argument shows that square BCHK has an area equal to rectangle BMLE.

Add: The two squares of the legs sum to the square of the hypotenuse!

But frequently the first proof one learns in early high school involves similar triangles...

- -- --- < Proof #2 > --- -- -

h is the altitude drawn to hypotenuse c. This creates a series of similar triangles drawn next to each other above.

• Using some corresponding sides from those similar triangles in the three diagrams, you get the following ratios, and can manipulate the equations as shown to get what we want:

^x/_a = ^a/_c, & ^y/_b = ^b/_c
x = ^a²/_c & y = ^b²/_c.
c = x + y = ^a²/_c + ^b²/_c = ^{a² + b²}/_c
a² + b² = c²

Even President James A. Garfield is credited with a proof:

- -- --- < Proof #3 > --- -- -

Given the trapezoid as shown, represent the area two different ways:
·A_trap = ½h(b₁+b₂) ½(a+b)(a+b) ;
½(a² + 2ab + b²).

·Breaking it down, Area = 2(½ab) + ½(c²);

½(a² + b²) + ab = ab + ½(c²);
a² + b² = c²

- -- --- < Proof #4 > --- -- -

This similar proof involves a square with side length = a + b; diagram below left. It is credited to the Indian mathematician Bhaskara, who lived c. 1150.

Here we do a similar proof to that above. Represent the area two ways:

(1) A_square = s² = (a + b)²
And breaking it down...
(2) Area = 4(½ab) + c²

a² + 2ab + b² = 2ab + c²
And yet again, a² + b² = c²

Number rules the universe.
Number is the within of all things.
Number is the ruler of forms and ideas,
and the cause of gods and demons.
Geometry is knowledge of the eternally existent.
There is geometry in the humming of the strings.

- -- --- Pythagoras --- -- -

Onto² = Part² + 2²