The Lux Problem

The Lux Problem
The following interesting geometry problem was recently (2020) brought to a class of a colleague and friend of mine, Hans Humenberger, from the University of Vienna, Austria by a student with the surname Lux - hence the title 'The Lux Problem' for this webpage:

Let c₁ and c₂ be two circles intersecting in A and B and a straight line through A is drawn, intersecting the two circles in M and N. Further let K be the midpoint of MN, P the intersection point of the angle bisector of ∠MAB with c₁, R the intersection point of the angle bisector of ∠ BAN with c₂. Prove that ∠ PKR = 90^o.

The student got the problem from his grandfather, a retired mathematics teacher from France, but without a solution. So the problem has been around for some time.

 

The Lux Problem

Challenge
1) Can you explain why (prove that) this result is true?
2) If you get stuck, press the Hint button in the dynamic sketch at the top. Then view Pompe's Hexagon Theorem.
3) Alternatively, have a look at this sketch for a hint at a dynamic proof of the result.
4) If still stuck, view two other different, but equivalent versions of the same problem, namely, Klingens' Theorem.

If still stuck for a solution, or to compare your own solution(s), read the joint paper of mine with colleague Hans Humenberger, University of Vienna, in the free online journal At Right Angles in the March 2021 issue at Ghosts of a Problem Past. This paper gives 4 different proofs as well as links to several other proofs, including one showing the result as a special case of a similar rectangle generalization of Van Aubel's quadrilateral theorem.

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
Klingens' theorem of two intersecting circles or two adjacent isosceles triangles (Equivalent variations of the Lux Problem)
Special Case of Van Aubel: Dick Klingen's Problem
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.)

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