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Triangle geometry with lasers

Source: Canada Repêchage 2014/4

June 18, 2016
geometry

Problem Statement

In △ABC\triangle ABC, the interior sides of which are mirrors, a laser is placed at point A1A_1 on side BCBC. A laser beam exits the point A1A_1, hits side ACAC at point B1B_1, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side ABAB at point C1C_1, then side BCBC at point A2A_2, then side ACAC again at point B2B_2, then side ABAB again at point C2C_2, then side BCBC again at point A3A_3, and finally, side ACAC again at point B3B_3.
(a) Prove that ∠B3A3C=∠B1A1C\angle B_3A_3C = \angle B_1A_1C.
(b) Prove that such a laser exists if and only if all the angles in △ABC\triangle ABC are less than 90∘90^{\circ}.
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