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=120o) 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.
This page was created by Michael de Villiers, 25 July 2009, updated 10 October 2020 with WebSketchpad.