Wednesday, March 19, 2008

Squares of Triangles

Pythagoras was a Greek philosopher and mathematician who lived in the sixth century B.C.E. He acquired a following in his time, and there was a cult built up around him in which he was worshiped as a demi god1. Today, he is most well known for his contribution to Euclidian geometry: the Pythagorean Theorem. Today's challenge is to prove the Pythagorean Theorem.


Given any right triangle ABC, prove A2 + B2 = C2 using only simple geometric formulas. (No trigonometric functions allowed).


This problem seeped into my head as I was trying to fall asleep some nights ago, and I never could solve it lying in bed in the dark. It wasn't until I sat down with a pencil and paper and studied the problem from several different angles (during the second and third quarters of the Superbowl) that I came up with a solution.

The first thing I did was to redraw the diagram like this:
That way, I could visualize the quantities, A2, B2 and C2 as the areas of the squares made from the sides of the triangles.

Next, I started to think about this:
Maybe we can find a relationship between a square with the side lengths of A + B and the grey square with lengths C.

The final thing you need to know is that the area of a triangle is equal to half the base times the height: The answer becomes clear when you draw in the rest of the triangles. Look at the red square.The lengths of each of its sides are equal to A + B, so it has the area (A + B)2.

If you look at the triangles shaded in should see that each has a height of A and a base of B, so their areas are ½AB. Therefore, the area of the grey square is equal to the area of the red square from above minus the areas of the four triangles.

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