Equi-inclined Lines Problem

NOTE: Please WAIT while the applet below loads.

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

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 by choosing X, Y and Z to respectively coincide with the vertices A, B and C.

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Created by Michael de Villiers, 24 April 2020.