Definition: An Apollonius quadrilateral is a quadrilateral for which the two products of its opposite sides are equal.
The following property of an Apollonius quadrilateral was recently discovered (April 2025) with the aid of dynamic geometry.
The Perpendicular Bisectors of an Apollonius Quadrilateral
The perpendicular bisectors of the sides of an Apollonius quadrilateral ABCD form another Apollonius quadrilateral A'B'C'D'.
Explore
Use the dynamic sketch below to explore this result by dragging.
Does the result also hold if ABCD becomes concave or crossed?
Perpendicular Bisectors of an Apollonius Quadrilateral
Challenge
1) Can you prove the result? Can you prove it in more than one way? Can you prove it purely geometrically? Is your proof general enough to cover all the cases above?
Hint: Click on the 'Link to PB-quad similarities' button to navigate to a new sketch showing a general quadrilateral ABCD and its corresponding perpendicular bisector (PB) quadrilateral. What do you notice about the four pairs of corresponding triangles formed by the diagonals of the two quadrilaterals? Can you use this observation as a Lemma to prove 1)?
Challenge: Can you prove the Lemma?
2) What about the converse? Is it also true? Prove or disprove it.
References
Papers on these results by Hans Humenberger, Austria, and myself has been accepted for publication in Mathematics in School. All rights reserved.
Humenberger, H. & De Villiers, M. (2026a). Perpendicular bisector quadrilaterals as a substantial learning environment. Mathematics in School, 55, 1, XX-YY.
Humenberger, H. & De Villiers, M. (2026b). Perpendicular Bisector Quadrilaterals of Parallelograms. Mathematics in School, 55, 2, XX-YY.
Harmonic Quadrilateral
An Apollonius quadrilateral that is cyclic is called a harmonic quadrilateral. In the case of a harmonic quadrilateral the perpendicular bisectors to the sides are concurrent and no quadrilateral is formed.
However, harmonic quadrilaterals also have some interesting properties and sometimes appear in problem solving contests - see for example, An extension of the IMO 2014 Problem 4 and Truong et al (2012).
Reference
Truong, P.N.V, Khanh, L.D. & Quang, B. H. D. (2012). About the harmonic quadrilateral. A paper presented at a mathematics conference in Bulgaria.
Related Links
Water Supply: Four Towns (Introduction to perpendicular bisectors - Rethinking Proof activity)
The Perpendicular Bisectors of a Parallelogram
Some Trapezoid (Trapezium) Explorations (See Investigation 5 about perpendicular bisectors)
IMO 2014 Problem 4 - Geometry
An extension of the IMO 2014 Problem 4
The quasi-circumcentre of a quadrilateral
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
Perpendicular Bisectors of Circumscribed/Tangential Quadrilateral
Conway’s Circle Theorem as special case of Windscreen Wiper Theorem
A 1999 British Mathematics Olympiad Problem and its dual
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
External Links
Apollonius quadrilateral (Wikipedia)
Harmonic quadrilateral (Wikipedia)
Perpendicular bisector construction of a quadrilateral (Wikipedia)
Quadrilaterals Formed by Perpendicular Bisectors (Cut The Knot)
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, 26-29 May 2025; updated 12 Nov 2025.