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

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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').

Also note that formulation 1 of the result is now 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.

Special case of Van Aubel generalization

Reference: De Villiers, M. (1998). Dual generalizations of Van Aubel’s theorem. The Mathematical Gazette, Nov, 405-412.

An interactive, online sketch of this 'similar rectangle' generalization of Van Aubel's theorem, as well as others can be found at Van Aubel’s theorem and some generalizations.

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Created 18 August 2013, by Michael de Villiers; modified 15 March 2020.