The laser game

In an equilateral triangular mirror room a laser is shot from the top vertex down to the base. The light ray bounces three times to return to the same base point. What is the angle α?

Scroll down for a solution to this problem.


The angle α is 30°.

Solution by Matthew Arcus.

Unfold reflections to five triangles, then left red triangle rotates to right about E by 120° and H is on the intersection with the circle through AEJ (angle at E and H is 120°). Then angle HAJ is 15° (half angle on AE which is 30°) so black angle is 105° and α = 2(105-90) = 30°.

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