Equi-inclined Lines Problem

Explore
From each vertex of ΔABC, draw an equi-inclined line to form an equal angle with the opposite side. What do you notice about the angles of the formed ΔHIG in relation to those of ΔABC? Explore more by dragging some of the points.

Equi-inclined Lines Problem

Challenge
1) You should've noticed that the corresponding angles are equal, and hence that ∆HIG is similar to ∆ABC. Can you explain why (prove that) the result is true?
2) Can you generalize further? For example: Can the equi-inclined lines lie in the extensions of sides AB, BC, and CA? Or does the result generalize to a quadrilateral or higher polygons? Explore!

Note: The above result can be viewed as a special case of Generalization 2 of a theorem by Wares given at: Further generalizations of the theorem of Wares. (The above result is obtained with Generalization 2 at the preceding link by choosing X, Y and Z to respectively coincide with the vertices A, B and C).

More Equi-inclined lines
Other examples of results involving equi-inclined lines:
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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