(A generalization of a theorem for 'equilic' quadrilaterals)

**Theorem**: If similar triangles *APC*, *DQC* and *DRB* are constructed on *AC*, *DC* and *DB* of any quadrilateral *ABCD* so that ∠*APC* = ∠*ASB*, where *S* is the intersection of *AD* and *BC* extended, then *P*, *Q*, *R*, and *S* are collinear.

(This result can be extended to cover the case when *AD* // *BC* by using the projective concept of 'vanishing' points, in which case the vanishing points are all collinear on the line at infinity.)

The above theorem is a generalization of the following result in Honsberger (1985), namely, "If *ABCD* is an 'equilic' quadrilateral (a quadrilateral with *AD* = *BC* and ∠*A* + ∠*B*=120^{o}) and equilateral triangles are drawn on *AC*, *DC* and *DB*, away from *AB*, then the three new vertices, *P*, *Q* and *R* are collinear."

Honsberger, R. (1985). *Mathematical Gems III. (Dolciani Mathematical Expositions)*. Washington, DC: MAA.

Quadrilateral Similar Triangles Collinearity Theorem

**Challenge**

1) Can you *explain why* (prove) the result is true?

2) If stuck, click on the *Hint* button. What do you notice? Try to use this observation together with similarity transformations to explan/prove the result.

A proof is given in De Villiers, M. The Role of Proof in Investigative, Computer-based Geometry: Some personal reflections. Chapter in Schattschneider, D. & King, J. (1997). *Geometry Turned On!* Washington: MAA, pp. 15-24. For more info go to *Amazon: Geometry Turned On*.

My 2009 book "Some Adventures in Euclidean Geometry" also contains a proof & the extension to the case when *AD* // *BC*, and is available as a PDF Download at Lulu. For more info go to *Some Adventures in Euclidean Geometry*.

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This page was created by Michael de Villiers, 25 July 2009, updated 10 October 2020 with *WebSketchpad*.