Sum of Quartic Distances to Vertices of an Equilateral Triangle
On 27 June 2026 in the ‘Romantics of Geometry’ group on Facebook, David Vreken from Marietta GA, USA, posted the following interesting problem:
'.....'If P is an arbitrary point on the incircle of a triangle ABC, then PA4 + PB4 + PC4 = 99r4 where r is the radius of the incircle.
Notes
i) This problem was earlier posted on Facebook in May 2017 by Van Khea from Cambodia.
ii) The reader is invited to explore the result dynamically in the first sketch on the left directly below.
iii) This problem obviously directly relates to the squared distance results given at Sum of Squared Distances related to Incircles & Circumcircles and Sum of Squared Distances to Vertices of Polygons.
Generalization
Following from a result mentioned by Van Khea on Facebook, the above result for 4th powers can be generalized as follows:
'.....'If P is an arbitrary point on a circle with radius r centered at O of a triangle ABC, then PA4 + PB4 + PC4 = 3(R4 + 4R2r2 + r4) where R is the radius of the circumcircle of ABC.
iv) The reader is invited to explore the result dynamically in the second sketch on the right directly below.
Challenge
Can you prove the results stated above? Can you prove them in more than one way?
(Compare your proof for the 1st problem with these selected Solutions posted on Facebook in June/July 2026).
Sum of powers of distances to vertices and sides
Further Exploration & Generalization
1) What happens if we similarly consider an arbitrary point P on a circle centered at O of any regular polygon, and investigate which sums of m-th powers of distances from P to the vertices would remain constant? What happens if we consider the same question for semi-regular side-gons or semi-regular angle-gons?
2) What happens if we investigate the sums of the m-th powers of distances from P to the sides instead of the vertices?
3) Make your own dynamic constructions to explore these questions, or work through the selected examples given below.
Powers of distances to vertices of a regular hexagon
4) Click on the 'Link to regular hexagon' button to navigate to a new sketch showing a circle centered at O of a regular hexagon with the distances from P to the vertices displayed and the sum of the m-th powers of these distances are calculated.
5) The dynamic sketch in 4) for a regular hexagon shows a value for m = 2. Drag P to confirm that the sum of the squared distances remains constant (as also illustrated at Sum of Squared Distances to Vertices of Polygons).
6a) Change the parameter m by clicking and changing it to other values. Check by dragging P to find out which sums of m-th powers of distances for a regular hexagon remain constant.
6b) Also change the size of the circle on which P lies by dragging X and check what happens by dragging P again.
7) Can you formulate a conjecture for any regular polygon? Can you prove your conjecture?
Powers of distances to vertices of a semi-regular side-hexagon
8) Click on the 'Link to semi-regular side-hexagon' button to navigate to a new sketch showing a circle centered at O of a semi-regular side-hexagon with the distances from P to the vertices displayed and the sum of the m-th powers of these distances are calculated.
9) The dynamic sketch in 8) for a semi-regular side-hexagon shows a value for m = 2. Drag P to confirm that the sum of the squared distances remains constant (as also illustrated at Sum of Squared Distances to Vertices of Polygons).
10a) Change the parameter m by clicking and changing it to other values. Check by dragging P to find out which sums of m-th powers of distances for a semi-regular side-hexagon remain constant.
10b) Also change the size of the circle on which P lies by dragging X and check what happens by dragging P again.
10c) Also change the shape of the semi-regular side-hexagon until it becomes concave & check whether your observations still hold.
11) Can you formulate a conjecture for any semiregular side-2n-gon? Can you prove your conjecture?
Powers of distances to vertices & sides of a semi-regular angle-hexagon
12) Click on the 'Link to semi-regular angle-hexagon' button to navigate to a new sketch showing a circle centered at O of a semi-regular angle-hexagon with:
a) the distances from P to the vertices displayed and the sum of the m-th powers of these distances are calculated.
b) the (perpendicular) distances from P to the sides displayed and the sum of the m-th powers of these distances are calculated.
13) The dynamic sketch in 12) for a semi-regular angle-hexagon shows a value for m = 2. Drag P to confirm that the sum of the squared distances to the vertices and sides remains constant. (The constant sum to the sides is also illustrated at Sum of Squared Distances related to Incircles & Circumcircles for the circumcircle of a semiregular angle-hexagon).
14a) Change the parameter m by clicking and changing it to other values. Check by dragging P to find out which sums of m-th powers of distances to vertices and sides for a semi-regular angle-hexagon remain constant.
(Note that the distances measured to the sides are displayed as 'directed distances' and these distances will change sign if P is dragged 'outside' corresponding sides. (More precisely, the distances to the sides will change sign, when their direction changes).
14b) Also change the size of the circle on which P lies by dragging X and check what happens by dragging P again.
14c) Also change the shape of the semi-regular angle-hexagon until it becomes crossed & check whether your observations still hold.
15) Can you formulate conjectures for any semiregular angle-2n-gon? Can you prove your conjectures?
Challenge
Can you prove all your observations in 7), 11) and 15)?
(Compare your proofs with Humenberger at al (2026) below).
References
Altshiller-Court, N. (2007). Theorem 109. College Geometry, Dover Publications, (1st published 1952, Barnes & Noble), pp. 70-71.
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. (2025). Unexpected Properties of Thales’ Circle, Revealed by Dynamic Geometry
Experiments. Mathematics in School, 54.4 (2025), pp. 14–17.
Humenberger, H.; Hungerbühler, N. & De Villiers, M. (2026). Constant Sums of Distances to Vertices and Sides of Polygons. International Journal of Geometry, Vol. 15, No. 4, pp. 73 - 87. ISSN: 2247-9880.
Meskhishvili, M. (2020). Cyclic Averages of Regular Polygons and Platonic Solids. Commun. Math.
Appl. 11.3 , pp. 335–355.
Meskhishvili, M. (2021). Cyclic averages of regular polygonal distances. Int. J. Geom. 10.1 (2021), pp. 58–65. issn: 2247-9880,2247-9880.
Related Links
Distances in an Equilateral Triangle (Viviani's Theorem) (Rethinking Proof activity)
2D Generalizations of Viviani's Theorem to Polygons
Airport Problem (Rethinking Proof activity, Fermat's minimization problem)
Sum of Squared Distances related to Incircles & Circumcircles
Sum of Squared Distances to Vertices of Polygons
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
Midpoint Trapezium Theorem
Kite Midpoints (Rethinking Proof activity)
Isosceles Trapezoid Midpoints (Rethinking Proof activity)
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)
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)
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
External Links
Sum of powers of distances between points in concentric circles (Post by Tommy at Mathematics Stack Exchange)
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, 14 August 2026; updated 17 August 2026; 2 Oct 2026.