Klingens' theorem of two intersecting circles or two adjacent isosceles triangles
(An application of a generalization of Van Aubel's Theorem)

In memoriam: A tribute to Dick Klingens (7 Mei 1945 - 24 Mei 2021)

The following intriguing problem was circulated by Dick Klingens at the annual meeting of the Nederlandse Vereniging van Wiskundeleraren (NVvW) in November 2011:
Formulation 1
Two circles K₁ and K₂ intersect each other in B and Q. A line through Q cuts K₁ in point A and K₂ in point C. The points E and F are the respective midpoints of the arcs BA and BC that do not contain Q. If M is the midpoint of AC, prove that ∠EMF = 90^o

Widgets

Dick Klingens' Problem

A different, but equivalent formulation of the same problem is the following:
Formulation 2
Given two isosceles triangles ABC and BDE with respective bases AB and BD lying in a straight line (and on the same side of the line), and BC is perpendicular to BE. If M is the midpoint of AD, prove that ∠CME = 90^o.

Important: To view & manipulate the dynamic version of this 2nd formulation, navigate to it using the appropriate button in the ABOVE dynamic sketch; the picture below is static.

Alternative, equivalent formulation

Challenge
Can you prove either one of these formulations? This extremely rich problem (but deceptively hard problem) can be proved in many different ways, which enhances its beauty and intrigue. If you get stuck, one possible hint is given in the second version above - press the Hint button in the dynamic sketch at the top.

Some Solutions
1) Compare your own solution to this one, Application of a generalization of Van Aubel (pdf), which includes a link to an interactive sketch as well as a link to a paper in Dutch with 5 different proofs of the result.
2) A download link to 9 more different proofs in Dutch in PDF format, zipped together in one file (3 MB), is available to download at More proofs.
3) In Sept 2013, my friend and colleague Poobhal Pillay, the Siyanqoba Math Problem Solving Development Coordinator for KwaZulu-Natal High Schools, supplied the following short, elegant proof of the 2nd version above using only the theorem of Pythagoras: Poobhal Proof (pdf).
4) The first formulation of the Klingens problem has also been proposed, slightly differently, on Alex Bogomolny's website in 2017 by Thanos Kalogerakis, Greece, and elegantly proved at: A Problem in Pentagon with Supplementary Angles.
5) More recently, in 2020, the first formulation of Dick Klingens' theorem resurfaced as the Lux Problem. It was brought to my attention by Hans Humenberger, Austria, and the preceding link to a dynamic sketch of the Lux problem, gives some new proofs as well as revisiting some old ones.

Dick Klingens'Homepage
Visit Dick Klingens' website at his Homepage (in Dutch) for a many more interesting, geometric results.

Some References
De Villiers, M. (1998). Dual Generalizations of Van Aubel's theorem. The Mathematical Gazette, Nov, pp. 405-412.
De Villiers, M. (2012). Another proof of De Opgave 2011: Application of a generalization of Van Aubel. Euclides, December, no. 3, p. 133.
De Villiers, M. & Humenberger, H. (2021). Ghosts of a Problem Past. At Right Angles, March, pp. 105-111.
Lecluse, T. (2012). Vanuit de oude doos: De Opgave 2011, uitgedeeld op de Jaarvergadering. Euclides, 87(5), Maart, no. 1, pp. 215-217.
Pillay, P. (2013). Proof of Klingen's Problem using Pythagoras. Personal communication.

Some Related Links
Special Case of Van Aubel: Dick Klingen's Problem
Lux Problem (Another variation of Dick Klingens)
Parallelogram Squares (Rethinking Proof activity)
Van Aubel's Theorem and some Generalizations
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
An associated result of the Van Aubel configuration and some generalizations
The Vertex Centroid of a Van Aubel Result involving Similar Quadrilaterals and its Further Generalization
Dào Thanh Oai's Perpendicular Lines Van Aubel Generalization
Some Corollaries to Van Aubel Generalizations
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
A Van Aubel like property of an Equidiagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
A Fundamental Theorem of Similarity
Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point
Pompe's Hexagon Theorem
Sum of Two Rotations Theorem
Some Variations of Vecten configurations
A Vecten area variation (Cross's theorem) & generalizations to quadrilaterals
Napoleon's Theorem (Rethinking Proof activity)
Napoleon's Theorem: Generalizations, Variations & Converses

Some External Links
A Problem in Pentagon with Supplementary Angles (Cut The Knot)
De stelling van Van Aubel en algemenisering daarvan (in Dutch)
Van Aubel's theorem (Wikipedia)
van Aubel's Theorem (Wolfram MathWorld)
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.)

********************************