A special case of a similar rectangle generalization of Van Aubel's theorem

The Klingens' theorem of "Two intersecting circles or two adjacent isosceles triangles" can be seen as a special case of the following generalization of Van Aubel’s theorem proved in De Villiers (1998): if similar rectangles E, F, G and H are constructed in alternating order (orientation) on the sides of any quadrilateral ABCD, then the lines connecting the centres of the rectangles on the opposite sides of ABCD are perpendicular to each other.

For example, consider the figure below, which shows sides CD and DA of the quadrilateral ABCD in a straight line. From this Van Aubel generalization, it then follows that EG and FH are perpendicular in the point O, which therefore also lies on the circle HDMG (see Application of a generalization of Van Aubel (pdf).

Also note that formulation 1 of Klingens' theorem is also quite nicely illustrated in the top part of the figure by the circles K₁ and K₂ intersecting in B and Q, the straight line AQC, where in this case we have EOMQF concyclic.

Widgets

Special case of Van Aubel similar rectangle generalization

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
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.)

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