Generalizations of a theorem by Wares1

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Generalization 1: Choose arbitrary points X, Y and Z. 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, then ∆UVW is similar to ∆XYZ. (1See A theorem of Wares.)

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Arbitrary points X, Y and Z

Generalization 2: If equi-inclined (making same angles) lines 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 equi-inclined lines as indicated, then ∆KLM is similar to ∆XYZ. (Point P can be dragged below to change the angle the equi-inclined lines make with the sides of XYZ.)

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Equi-inclined lines to sides of XYZ

Generalization 3: Choose arbitrary points E, F, G and H. If perpendiculars are dropped from A, B, C and D to line segments HE, EF, FG and GH respectively, and if P, Q, R and S are the intersections of these perpendiculars as indicated below, then PQRS is equi-angled to EFGH.

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Generalization to quadrilateral

Regarding the above, read my paper in The Scottish Mathematical Council Journal (2012), Generalizing a theorem of Arsalan Wares.


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By Michael de Villiers. Created, 1 April 2014.