Trigonometry/Circles and Triangles/Viviani's theorem

Viviani's theorem is as follows: Take any point P in an equilateral triangle. Drop perpendiculars from P to the three sides; let their feet be D, E, F. Then whatever the location of P, PD+PE+PF equals the altitude of the triangle. If P is outside the triangle, this still holds provided that the perpendicular or perpendiculars lying entirely outside the triangle are treated as having negative length.