## 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.

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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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By Michael de Villiers. Created, 1
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