Viviani in Isosceles Triangle: What is it?
A Mathematical Droodle

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Explanation

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This statement leads directly to Viviani's theorem:

The sum of distances of a point inside an equilateral triangle or on one of its sides equals the length of its altitude.

With the reference to the diagram below,

Viviani's theorem asserts that the sum PS + PT + PU is indendent of P.

Draw B_aC_a||BC:

As we just proved, in the isosceles ΔB_aAC_a, the sum PS + PT is independent of the position of P on B_aC_a. Since B_aC_a||BC, this is also true of PU. Therefore, for P∈B_aC_a, the sum PS + PT + PU is independent of the position of P.

Since ΔABC is equilateral the same holds for P on A_bC_b||AC:

It follows that the sum PS + PT + PU is independent of the position of P on the union of the two lines. Now, when P glides over B_aC_a, A_bC_b sweeps the whole of ΔABC, showing that the sum remains the same for any position of P inside ΔABC, thus proving Viviani's theorem.

For another proof, simply observe that

Area(ΔABP) + Area(ΔACP) = Area(ΔABC).

Writing that explicitly we obtain

AB·PS/2 + AC·PT/2 = AC·BT'/2,

and, taking into account that AB = AC,

PS + PT = BT' = const.

Related material Read more...
	Equilateral and 3-4-5 Triangles
	Rusty Compass Construction of Equilateral Triangle
	Equilateral Triangle on Parallel Lines
	When a Triangle is Equilateral?
	Viviani's Theorem
	Viviani's Theorem (PWW)
	Viviani by Vectors
	Morley's Miracle
	Triangle Classification
	Napoleon's Theorem
	Sum of Squares in Equilateral Triangle
	A Property of Equiangular Polygons
	Fixed Point in Isosceles and Equilateral Triangles

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