Generalizations of a theorem by Wares

"In mathematics it is often the case that a new result is obtained by looking at an old result in a new way and extending it to obtain a generalization of the old result. Hence it frequently happens that: Today’s theorem becomes tomorrow’s corollary." - Gary Chartrand, Albert Polimeni & Ping Zhang (2003: 107)

Below some different generalizations of a Theorem by Wares are given.

Generalization 1
Given a ∆ABC 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.

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
Given a quadrilateral ABCD 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

Further Generalization
Can you generalize the above results further to other polygons?

References
Chartrand, G; Polimeni, A. & Zhang, P. (2003). Mathematical Proofs: A transition to advanced mathematics. Boston: Addison Wesley.
De Villiers, M. (2012). Generalizing a theorem of Arsalan Wares. The Scottish Mathematical Council Journal, No. 42, pp. 41-43.

More Equi-inclined lines
Other examples of results involving equi-inclined lines:
Equi-inclined Lines Problem
Equi-inclined Lines Generalizations of Viviani's Theorem
A generalization of Neuberg's Theorem and the Simson line
A variation of Miquel and its generalization
Generalizations of a theorem by Wares
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
Rigby's Eight Point Conic for a Quadrilateral

Related Links
A theorem by Wares
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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