A Proof for the Converse of the Pythagorean Theorem

Converse of the Pythagorean Theorem:

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Proof:
Suppose the triangle is not a right triangle. Label the vertices A, B and C as pictured. (There are two possibilities for the measure of angle C: less than 90 degrees (left picture) or greater than 90 degrees (right picture).)

Construct a perpendicular line segment CD as pictured below.

By the Pythagorean Theorem, BD² = a² + b² = c², and so BD = c. Thus we have isosceles triangles ACD and ABD. It follows that we have congruent angles CDA = CAD and BDA = DAB. But this contradicts the apparent inequalities (see picture) BDA < CDA = CAD < DAB (left picture) or DAB < CAD = CDA < BDA (right picture).

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