Is this valid for a non-right isosceles triangle?

Geometry Level 2

It is well known that, in a billiard table shaped like a right isosceles triangle as in the diagram below, a ball struck at the marked vertex cannot return to that same point after any number of reflections. The ball may bounce any number of times, and if the ball hits a vertex point as opposed to a side, then the reflections are absorbed.

Does this also hold true for non-right isosceles triangles?

Yes No

2 solutions

Conrad Crowley
Jan 16, 2018

The answer is No. This can easily be seen by considering an equilateral triangle and striking the ball straight off the opposite side.

Well, it depends if you’re asking if this property is true for all non-such triangles, which doesn’t make much sense, because if I can choose the starting vertex as your example would suggest, then even right isosceles triangles fail because I can pick the right angled vertex and shoot from there. If, instead, you’re asking if there is another triangle or family of triangles with this property, then the answer is yes, for example if ABC is a triangle with angles <BAC = x and <ACB = n*x, n positive integer, then a ball shot from A will never return to its starting point.

Bernardo Picão - 2 years, 4 months ago

Manurag M
May 6, 2018

For the ball to come back it should retrace its path ie. it should hit a side perpendicularly. .
For a non right isosceles triangle it is possible in one go as it has an altitude within the area of the triangle and its intersection within the base(not at vertex).

Is this valid for a non-right isosceles triangle?

2 solutions

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