Sum of Squared Distances to Vertices of Polygons

Sum of Squared Distances to Vertices of any Triangle
Altshiller-Court (2007, pp. 70-71) proved the following remarkable result: For any point M in the plane of the triangle ABC the following relationship holds AM² + BM² + CM² = AG² + BG² + CG² + 3MG², where G is the centroid of ΔABC. But as pointed out by Bogomolny (2018), an interesting and useful corollary that follows from this relationship is that if P is an arbitrary point lying on any circle with its centre located at centroid G of a triangle, then the sum of squares of the distances from P to the vertices is constant.

Sum of Squared Distances to Vertices of an Equilateral Triangle
Furthermore, as also mentioned by Bogomolny (2018), since the incentre I and circumcentre O of an equilateral triangle both coincide with its centroid G, we immediately have that if P is an arbitrary point lying on the incircle or circumcircle of an equilateral triangle, then the sum of squares of the distances from P to its vertices is constant.

Dynamic Illustration
The dynamic sketch below illustrates these two results.
1. Drag P in the dynamic sketch below and observe how the measured distances change.
2. Click on the 'Show Sum Squared Distances' buttons in the figures below, and again drag P. What do you notice?
3. To view the 2nd result above, click on the 'Link to equilateral Δ's' button to navigate to a new sketch.
4. Drag P or R in the new sketch and observe how the measured distances change.
5. Click on the 'Show Sum Squared Distances' buttons in each of the figures in the new sketch, and again drag P and R. What do you notice?

Sum of Squared Distances to Vertices of Polygons

Sum of Squared Distances to Vertices: Further Generalizations
Using vectors and the arithmetic mean (average) definition of a vertex centroid, it is relatively easy to see and prove the following two further generalizations a) and b):
a) If P is an arbitrary point lying on any circle with its centre located at the vertex centroid G of a polygon, then the sum of squares of the distances from P to its vertices is constant.
6. Click on the 'Link to quadrilateral general' button to navigate to a new sketch illustrating generalization a) for a general quadrilateral. (Also check by dragging if the result holds for a concave or crossed ABCD).
b) If P is an arbitrary point lying on the incircle of a tangential polygon or the circumcircle of a cyclic polygon, and their respective incentres and circumcentres coincide with the vertex centroid G of the polygon, then the sum of squares of the distances from P to its vertices is constant.
7. Click on the 'Link to semi-regular quads' button to navigate to a new sketch which illustrates generalization b) for tangential and cyclic quadrilaterals. Explore the new sketch and formulate your observations.
8a. With reference to 7), note as shown in Humenberger & De Villiers (2026) that a rhombus is not the only tangential quadrilateral for which its respective incentre coincides with its vertex centroid: for example, a tangential isosceles trapezoid has its incentre coinciding with its vertex centroid, and there is also a particular kite with its incentre coinciding with its vertex centroid. Click on the 'Link to kite I = G' and 'Link to bicentric isosceles trapezium' buttons to navigate to new sketches illustrating these two tangential cases.
8b. However, note that a rectangle as shown in Humenberger & De Villiers (2026) is the only cyclic quadrilateral for which its circumcentre coincides with its vertex centroid.
Challenge
9. Can you further generalize the two results to pentagons, hexagons, etc.?
10a. Click on the 'Link to pentagons' and 'Link to semi-regular hexagons' buttons to navigate to new sketches. Explore the new sketches and formulate your observations.
10b. Can you find other tangential or cyclic pentagons and hexagons for which their respective incentres and circumcentres coincide with their vertex centroids?
Challenge
11. Can you prove your observations in 10? Or better still, provide general proofs to cover all possible cases for tangential or cyclic polygons with an odd or even number of vertices with their centres coinciding with their vertex centroids?

Application
12. The above result for tangential or cyclic polygons with their respective incentres and circumcentres coinciding with their vertex centroids provides an easy proof of the 1st case (and its generalization) at: Sum of Squared Distances related to Incircles & Circumcircles. However, note that the general result does not provide a proof of the 2nd case on the preceding webpage, since the 2nd case deals with the 'sum of the squared distances from a point Q on the circumcircle to the sides of the cyclic polygon'.

Sum of powers of distances to vertices and sides
An even further generalization to the sums of powers of distances from a point on a circle to the vertices and sides of regular and semi-regular polygons are explored at: Sum of powers of distances to vertices and sides.

References
Altshiller-Court, N. (2007). Theorem 109. College Geometry, Dover Publications, (1st published 1952, Barnes & Noble), pp. 70-71.
Bogomolny, A. (2018). Sum of Squares of Distances to Vertices. Webpage at Cut The Knot.
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.

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, and more, has been submitted for publication. All rights reserved.

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)
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, 31 May 2026; updated 2 June 2026; 14 August 2026; 19 August 2026.