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.)
Note: The above theorem is a generalization of the following 'geometrical gem' 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."
Quadrilateral Similar Triangles Collinearity Theorem
Challenge
1) Can you explain why (prove that) 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 explain/prove the result.
Note: De Villiers (1994; 2009) contains a proof as well as an extension to the case when AD // BC. A proof is also given in De Villiers (1997).
References
De Villiers, M. (1994; 1996; 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Press: Dynamic Mathematics Learning, pp. 93-97.
De Villiers, M. (1997). The Role of Proof in Investigative, Computer-based Geometry: Some personal reflections. Chapter in Schattschneider, D. & King, J. (1997). Amazon: Geometry Turned On!. Washington: MAA, pp. 15-24.
Honsberger, R. (1985). Mathematical Gems III. (Dolciani Mathematical Expositions). Washington, DC: MAA.
Some Related Links
Euler line proof
Euler-Nagel line Analogy
Four Collinear Points
Triangle Incentre-Circumcentre Collinearity
An interesting collinearity (An application of Desargues & Pappus)
Concurrency, collinearity and other properties of a particular hexagon
Opposite Side Quadrilateral Properties by Kalogerakis
Another Property of an Opposite Side Quadrilateral
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Van Aubel's Theorem and some Generalizations
Matric Exam Geometry Problem - 1949 (A variation of Reim's theorem)
More Properties of a Bisect-diagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
A Van Aubel like property of an Equidiagonal Quadrilateral
Pitot's theorem about a Tangential Quadrilateral
Extangential Quadrilateral
Theorem of Gusić & Mladinić
Constructing a general Bicentric Quadrilateral
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Japanese Circumscribed Quadrilateral Theorem
A Forgotten Similarity Theorem?
Some External Related Links
Equilic Quadrilateral (Wolfram MathWorld)
Quadrilateral (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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This page was created by Michael de Villiers, with JavaSketchpad 25 July 2009, updated 10 October 2020 to WebSketchpad; updated 10 Dec 2024; 25 March 2026.