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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 *K*_{1} and *K*_{2} 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.