Viviani's theorem

Viviani's theorem is that for a given point inside an equilateral triangle, the sum of the perpendicular distances from the point to the sides is equal to the height of the triangle. If the point is outside the triangle, the relationship still holds if one or more of the perpendiculars is treated as a negative value. Viviani's theorem generalizes to a regular n-sided polygon: the sum of the perpendicular distances from an interior point to the n sides being n times the apothem of the figure. The theorem is named for Vincenzo Viviani (1622–1703), a pupil of Galileo and Torricelli, who is also remembered for a reconstruction of a book on the conic sections of Apollonius and for finding a way of trisecting an angle through the use an equilateral hyperbola.