The following three results further generalize the results given at Some generalizations of an associated result of the Van Aubel configuration:

1) If a pair of directly similar triangles *BPA* and *DRC* are constructed on opposite sides *AB* and *CD* of quadrilateral *ABCD*, and another pair of directly similar triangles *ASD* and *CQB* are constructed on opposite sides *AD* and *CB* so that ∠*ASD* = ∠*BPA*, and if *F*, *G*, *H*, *I* are the midpoints of the segments *AC*, *BD*, *QS*, *PR* respectively, then *GHFI* is a quadrilateral with a pair of equal opposite angles at vertices *H* and *I* (see the Figure below)

2) If a pair of directly similar isosceles triangles *BPA* and *DRC* are constructed on opposite sides *AB* and *CD* of quadrilateral *ABCD*, and another pair of directly similar isosceles triangles *ASD* and *CQB* are constructed on opposite sides *AD* and *CB*, and if *F*, *G*, *H*, *I* are the midpoints of the segments *AC*, *BD*, *QS*, *PR* respectively, then *GHFI* is a kite with *HI* its axis of symmetry (navigate to the appropriate figure by clicking on the '**Link to Similar Isosceles Triangles**' button in the Figure below)

3) If a pair of directly similar triangles *BPA* and *DRC* are constructed on opposite sides *AB* and *CD* of quadrilateral *ABCD*, and another pair of directly similar triangles *ASD* and *CQB* are constructed on opposite sides *AD* and *CB* so that ∠*ASD* = 180° - ∠*BPA*, and if *F*, *G*, *H*, *I* are the midpoints of the segments *AC*, *BD*, *QS*, *PR* respectively, then *GHFI* is a cyclic quadrilateral (navigate to the appropriate figure by clicking on the '**Link to Apex Angles Supplementary**' button in the Figure below).

Van Aubel Associated Similar Triangles Further Generalizations

**Published paper**: My paper in the *Int. Journal of Math Ed in Sci & Technol.* discussing these results has been published online. The first 50 copies are free to download at: An associated result of the Van Aubel configuration and its generalization.

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Created with *WebSketchpad* 30 May 2021 by Michael de Villiers; updated 19 February 2022.