Sum of Squared Distances Duality
The following two problems in De Villiers (1994; 2009; pp. 67 & 172; 1995) display an interesting side-angle (vertex) duality:
'.....i) Prove that if P is any arbitrary point on the incircle of an equilateral △ABC, then PA2 + PB2 + PC2 is a constant.
'.....ii) Prove that if Q is any arbitrary point on the circumcircle of an equilateral △ABC, then h2AB + h2BC + h2AC is a constant (where hi are the respective distances from Q to the three sides).
Dynamic Illustration
The dynamic sketch below illustrates the two results.
1. Drag P or Q in the dynamic sketches below and observe how the measured distances change.
2. Click on the 'Show Sum Squared Distances' buttons in each of the figures below, and again drag P and Q. What do you notice?
Sum of Squared Distances related to Incircles & Circumcircles
Challenge
3. Can you prove the two results? Can you prove the two results in more than one way?
(Compare your proofs in 3) with those given in De Villiers (1994; 2009; pp. 194-195).
Further Generalization
4. Can you further generalize the two results to quadrilaterals?
5. Click on the 'Link to generalization 1' button to navigate to a new sketch. Explore the new sketch and formulate your observations.
Challenge
6. Can you prove your observations in 5?
(Compare your proofs in 6) with those given in De Villiers (1994; 2009; pp. 200-201).
7. Can you further generalize the two results to pentagons, hexagons, etc.?
8. Click on the 'Link to generalization 2' and 'Link to generalization 3' buttons to navigate to new sketches. Explore the new sketches and formulate your observations.
Challenge
9. Can you prove your observations in 8? Or better still, provide general proofs to cover all possible cases for polygons with an odd or even number of vertices?
(Hint: For proving the 1st case and its generalization above, you might find this general result helpful: Sum of Squared Distances to Vertices of Polygons.)
Converse Results
10. Formulate converse(s) of the above results for tangential and cyclic quadrilaterals and investigate whether they are also true. If so, prove them. If not, provide counter-examples.
Comment
As shown above, the two 'sum of squared distances' results for equilateral triangles generalize, respectively, to equilateral tangential polygons and equi-angular cyclic polygons. When these polygons have an odd number of sides, we obtain regular polygons, but if these polygons have an even number of sides, we respectively obtain semi-regular side-gons or semi-regular angle-gons as shown at: Semi-regular Angle-gons and Side-gons. (For proofs of these properties, see De Villiers, 2011).
Connection to Viviani's theorem
The 2nd configuration at the top directly relates to Viviani's theorem, which states that the sum of the distances from an arbitrary point Q to the sides of an equilateral triangle is constant - see for example this activity: Viviani's Theorem. Therefore, provided we use directed or signed distances, we also have in the 2nd case in the dynamic sketch above that hAB + hBC + hAC is a constant (where hi, are the respective distances from Q to the three sides). Since Viviani's theorem generalizes to equilateral polygons, equi-angled polygons or 2n-gons with opposite sides parallel - for example, see Generalizations of Viviani's Theorem to Polygons - the constant sum of the distances to the sides from Q to the sides also holds in generalizations 1-3, etc.
Submitted Paper
The following paper by Hans Humenberger and myself, namely, "Constant Sums of Squared Distances to Vertices and Sides of Polygons", regarding the dynamic generalizations illustrated here, has been submitted for publication. All rights reserved.
References
Altshiller-Court, N. (2007). Theorem 109. College Geometry, Dover Publications, (1st published 1952, Barnes & Noble), pp. 70-71.
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry. (Free to download). Morrisville, NC: Lulu Publishers, Dynamic Mathematics Learning.
De Villiers, M. (1995). Conjecture and Proof with Sketchpad: A case study. Pythagoras, 37 (August), pp. 30-32.
De Villiers, M. (2011). Equi-angled cyclic and equilateral circumscribed polygons. The Mathematical Gazette, 95(532), pp. 102-106.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning & Teaching Mathematics, No. 29, pp. 30-34.
Humenberger, H. & De Villiers, M. (2026). Investigating Centres of Cyclic and Tangential Quadrilaterals Coinciding with Centroids. IJMEST, July 2026, DOI:10.1080/0020739X.2026.2686127.
Related Links
Sum of Squared Distances to Vertices of Polygons
Sum of powers of distances to vertices and sides
Semi-regular Angle-gons and Side-gons (Generalizations of rectangles & rhombi)
A Generalization of the British Flag theorem to Semi-Regular Angle-2n-gons
Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons (Generalizations of isosceles trapezia & kites)
Investigating Centres of Isosceles Trapezia & Kites coinciding with their Centroids
Cyclic Quadrilateral Alternate Angle Sum (Rethinking Proof activity, incl. generalization to cyclic 2n-gons)
Logical Discovery: Tangential quadrilateral (Rethinking Proof activity, Pitot's theorem incl. generalization to tangential 2n-gons)
A Rectangle Angle Trisection Result (Application of symmetry)
A Rhombus Angle Trisection Result (Application of symmetry)
Distances in an Equilateral Triangle (Viviani's Theorem) (Rethinking Proof activity)
2D Generalizations of Viviani's Theorem to Polygons
Clough's Theorem and some Generalizations (A variation of Viviani)
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
Parallelogram Distances (An intro to directed distances)
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Angle Divider Theorem for a Cyclic Quadrilateral
Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
Parallel Lines (Thomsen's Hexagon) (Rethinking Proof activity; incl. of generalization)
External Links
Sum of Squares in Equilateral Triangle (Cut The Knot)
Sum of Squares of Distances to Vertices (Cut The Knot)
Sum of Squares of Distances to Vertices with GXWeb (YouTube video)
Sum of Squares Surprising Result - Out the Box Solution (YouTube video)
Viviani's theorem (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 25 May 2026; updated 28 May 2026; 30 May 2026; 2 June 2026; 19 August 2026.