# 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.