A theorem by Wares

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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?

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

1) 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?

2) 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.

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Created by Michael de Villiers, 1 April 2014. Modified 24 April 2020.