A theorem by Wares
The following result is from Wares (2010): Choose arbitrary points X, Y and Z, respectively on sides BC, CA and AB of ∆ABC. If perpendiculars are dropped from A, B and C to line segments YZ, XZ and XY, respectively, and if U, V and W are the intersections of these perpendiculars as indicated, what do you notice about the corresponding angles of ∆UVW and those of ∆XYZ? What can you conclude about these two triangles?
1) Explore your observations below by dragging. Do they hold when X, Y and Z are dragged on to the extensions of the sides?
Reference
Wares, A. (2010). Using dynamic geometry to explore non-traditional theorems. International Journal for Mathematical Education in Science & Technology, 41(3), pp. 351–358.
A theorem by Wares
Challenge
2) You should've noticed that ∆UVW is similar to ∆XYZ, since corresponding angles are equal. Can you explain why (prove that) the result is true? Can you prove it in more than one way?
Further Generalization
3) Reflect on your proof. Can you generalize further? For example: Is it necessary for points X, Y and Z to be on the sides of ABC? Is it necessary to draw perpendiculars to the sides of XYZ? Does the result generalize to a quadrilateral or higher polygons?
(Think about it for a while, and/or use dynamic geometry software to explore, before going to Further generalizations of the theorem of Wares and/or De Villiers (2012).
Reference
De Villiers, M. (2012). Generalizing a theorem of Arsalan Wares. The Scottish Mathematical Council Journal, No. 42, pp. 41-43.
Related Links
Generalizations of a theorem by Wares
Equi-inclined Lines Problem
Napoleon's Theorem (Rethinking Proof activity)
Miquel's Theorem (Rethinking Proof activity)
Triangle Altitudes (Concurrency, Rethinking Proof activity)
A Forgotten Similarity Theorem?
Perpendicular Trisectors of a Triangle & its generalization
The Perpendicular Bisectors of a Parallelogram
Some Trapezoid (Trapezium) Explorations (See Investigation 5)
Similar Parallelograms: A Generalization of a Golden Rectangle property
Some Variations of Vecten configurations
An associated result of the Van Aubel configuration and some generalizations
External Links
Arsalan Wares (Google Scholar)
Similarity (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
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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Created by Michael de Villiers, 1 April 2014 with JavaSketchpad, converted to WebSketchpad 24 April 2020, updated 18 Sept 2026.