## Some further generalizations of an associated result of the Van Aubel configuration using pairs of similar triangles

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).

#### Web Sketchpad .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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.