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


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Explanation

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Viviani in Isosceles Triangle

The applet attempts to illustrate the following proposition:

The sum of distances of a point in the base of an isosceles triangle to the sides does not depend on the position of the point on the base.


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For the proof, let PA'||AC, with A'∈AB. Draw BT'⊥AC and denote S' the intersection of BT' and BS'. Then, since BA'P is isosceles, PS = BS'. And, since S'T'TP is a rectangle, PT = S'T', so that

PS + PT = BS' + S'T' = BT',

the altitude from B to side AC, which is independent of the position of P.

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,

Configuration of Viviani's theorem

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

Draw BaCa||BC:

Configuration of Viviani's theorem. Step 1 in the proof

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

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

Configuration of Viviani's theorem. Step 2 in the proof

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 BaCa, AbCb 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
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  • Equilateral and 3-4-5 Triangles
  • Rusty Compass Construction of Equilateral Triangle
  • Equilateral Triangle on Parallel Lines
  • Equilateral Triangle on Parallel Lines II
  • When a Triangle is Equilateral?
  • Viviani's Theorem
  • Viviani's Theorem (PWW)
  • Tony Foster's Proof of Viviani's Theorem
  • Viviani by Vectors
  • Slanted Viviani
  • Slanted Viviani, PWW
  • 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
  • Parallels through the Vertices of Equilateral Triangle
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