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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*. (^{1}See *A theorem of Wares*.)

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 *K*, *L* and *M* are the intersections of these equi-inclined lines as indicated, then ∆*KLM* is similar to ∆*XYZ*. (Point *P* can be dragged in the linked sketch above to change the angle the equi-inclined lines make with the sides of *XYZ*.)

**Note**: The Equi-inclined Lines Problem in the *Student Explore* section can be viewed as a special case of the above generalization 2, and is immediately obtained by moving each of *X*, *Y* and *Z* respectively to a vertex of the original triangle.

**Important**: To view & manipulate the *dynamic version* of this generalization, and the next one, navigate to them using the appropriate buttons in the **ABOVE** dynamic sketch; the two pictures below are static.

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*.

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