Throwback Fact of the Week - Viviani's Theorem - 10/09/14

Viviani's Theorem can be broken into 2 steps:

1. Choose a point anywhere inside of an equilateral triangle
2. Draw perpendicular lines from the point to each of the 3 sides of the triangle

The theorem states that the sum of the lengths of these lines is equal to the height of the triangle. Using the image below, the theorem states that x + y + z = h, no matter where inside the triangle you place point P. 

The theorem was proven by Vincenzo Viviani around the year 1659. The proof can be derived easily from the formula for the area of a triangle (Area = .5bh, where b is the base of the triangle and h is the height). To quote the typical math textbook, "this proof is left as an exercise for the reader."

This theorem can be generalized to any regular n-sided polygon. For the case of an n-sided polygon, the sum of the perpendicular distances from an interior point to each of the n sides is equal to n times the length of the apothem of the polygon (Recall that an apothem of a regular polygon is a perpendicular line segment from the center of the polygon to the midpoint of one of its sides).