5 - Pleasing Proofs

Summary

Real mathematics … must be justified as art if it can be justified at all.

—G. H. Hardy (1877–1947), A Mathematician's Apology

In mathematics, assertions can be proved, which distinguishes mathematics from other disciplines. Mathematical knowledge is thus absolute and universal, independent of space and time. In this chapter, we present some proofs that are particularly memorable. Most are not well known and deserve to be better known.

The Pythagorean Theorem

The Pythagorean theorem states that given a right triangle, the area of a square formed on the hypotenuse is equal to the sum of the areas of the squares formed on the two legs.

There are many proofs of this important theorem. Figure 5.1 shows a tessellation proof. The plane is tessellated, or tiled, with copies of the square on the hypotenuse of the triangle (shaded in the figure), and also tessellated by copies of the squares on the two legs. This shows that the square on the hypotenuse can be divided into five pieces that can be reassembled to form the squares on the two legs. Two pieces make the smaller square and three pieces make the larger square.

The Erdős–Mordell Inequality

In 1935 Paul Erdős conjectured a geometric inequality. Let ABC be a triangle and M be a point in the interior or on the boundary of ABC. Let the distances from M to the vertices A, B, C be x, y, z, respectively, and let the distances from M to the sides AB, BC, CA be c, a, b, respectively.