Trigonometry/Circles and Triangles/Viviani's theorem

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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.