Sum of Squares in Equilateral Triangle

Here's problem #86 from a charming collection by C. W. Trigg, the long term editor of the Problems sections at the Mathematics Magazine.

If ABC is an equilateral triangle, and P is any point on the incircle of ΔABC, prove that   AP² + BP² + CP² is constant.

Solution

Reference

1. C. W. Trigg, Mathematical Quickies, Dover, 1985

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Copyright © 1996-2015 Alexander Bogomolny

If ABC is an equilateral triangle, and P is any point on the incircle of ΔABC, prove that   AP² + BP² + CP² is constant.

Solution

Trigg credits Leo Moser with the following solution.

Think of the whole diagram as being drawn in three dimensions, so that A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1), making the three points lie in the plane x + y + z = α (in fact, α = 1, in this case, but this is convenient to think of it as a generic constant for the sake of a lurking generalization.) The incircle of ΔABC is the intersection of that plane with the sphere x² + y² + z² = β, another constant.

For P on the incircle,

	AP² + BP² + CP²	= (1 - x)² + y² + z² +    x² + (1 - y)² + z +    x² + y² + (1 - z)²
		= 3(x² + y² + z²) - 2(x + y + z) + 3
		= 3β - 2α + 3
		= constant.

Observe that the proof goes through for any circle concentric to the incircle. A more general result claims that the sum of squares of the distances from a point to the vertices of a triangle - not necessarily equilateral - is constant for every circle with center at the centroid of the triangle.

2D Problems That Benefit from a 3D Outlook

1. Four Travellers, Solution
2. Desargues' Theorem
3. Soddy Circles and Eppstein's Points
4. Symmetries in a Triangle
5. Three Circles and Common Chords
6. Three Circles and Common Tangents
7. Three Equal Circles
8. Menelaus from 3D
9. Stereographic Projection and Inversion
10. Stereographic Projection and Radical Axes
11. Sum of Squares in Equilateral Triangle

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	Problem in Equilateral Triangle
	Problem in Equilateral Triangle II
	Triangle Classification
	Isoperimetric Property of Equilateral Triangles
	Maximum Area Property of Equilateral Triangles
	Angle Trisectors on Circumcircle
	Equilateral Triangles On Sides of a Parallelogram
	Pompeiu's Theorem
	Circle of Apollonius in Equilateral Triangle

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Copyright © 1996-2015 Alexander Bogomolny