The Converse of Viviani's Theorem

Abstract

Viviani's Theorem, discovered over 300 years ago, states that inside an equilateral triangle, the sum of the perpendicular distances from a point P to the three sides is in dependent of the position of P (and so equals the altitude of the triangle). Incidentally, the person for whom this theorem is named is Vicenzo Viviani (1622-1703), a pupil of both Galileo and Torricelli. The theorem can easily be proved: Let s and h be the side length and the altitude of the equilateral triangle AABC, let P be any point inside the triangle, and let du d2, and d3 be the three distances from P to the sides of the triangle. Since AABC is made up of APAB, APBC, and APCA, it follows that \sh = \sdx + \sd2 + \sd3, and so d + d2 + d$ = h, which proves the theorem. Samelson [1] gave a different proof, one that uses vectors, something we will return to shortly. This motivated us to consider the following question: Does the converse of Viviani's Theorem also hold? That is, if the sum of the distances from a point inside a triangle to the three sides is constant, must the triangle be equilateral? An affirmative answer can be obtained just by considering different points near one vertex of the triangle. However, by using vectors (in the style of Samelson), we can get a stronger result.