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
De Villiers, M. (2023). An associated result of the Van Aubel configuration and its generalization. International Journal of Mathematical Education in Science and Technology, Vol 54, no 3, pp. 462-472.
Related Links
An associated result of the Van Aubel configuration and some generalizations
An associated result of the Van Aubel configuration and its generalization: Different Quadrilateral Arrangements
A Fundamental Theorem of Similarity
Van Aubel Centroid & its Generalization
Van Aubel's Theorem and some Generalizations
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Dào Thanh Oai's Perpendicular Lines Van Aubel Generalization
Generalizations of a theorem of Sylvester (about forces acting on a point in a triangle)
Napoleon's Theorem: Generalizations & Converses
Some Variations of Vecten configurations
Parallelogram Squares (Rethinking Proof activity)
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Created with WebSketchpad 30 May 2021 by Michael de Villiers; updated 19 February 2022; 13 August 2025.