Pythagorean Theorem
Greg Huberty

The Pythagorean theorem has had several different proofs many of which are very similar. On that I found to be very different and interesting is the proof that the former president of the United states James Garfield is credited with finding. It is not like other proofs in that it does not use squares of the sides, rather it uses two triangles of sides a, b, and c along with a triangle with legs of c to create a trapezoid. Here is the proof

First, let us make the trapezoid. You start with a triangle of sides a, b, and c where the sides a and b meet to form a right angle.

Then put a second triangle below the first such that side a is an extension of the other triangles b side.

Next connect the end of side a at the top with side b on the bottom to create the trapezoid.

Remembering the formula for the area of a trapezoid to be
the formula for the area of this trapezoid will be . This area will be the same as the sum of the areas of the three triangles that are located inside the trapezoid

Setting these equal you get
Simplifying both sides you come to
By multiplying both sides by two it becomes

Then subtracting both sides by 2ab the end result is which is the Pythagorean Theorem.

Another proof of this theorem is the DaVinci proof of using areas and cutting and pasting them to show equal areas. Click here.

For more ways of looking at the Pythagorean Theorem, try these:

Behold Pythagoras
Leonardo da Pythagoras
Puzzled Pythagoras
Shear Pythagoras

This is another place to look to see more about the Pythagorean Theorem. Click here.

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