This week, we have a sort of geometry problem dealing with billiards. It comes in two parts.
A standard billiards table is a rectangle with no "pockets." The sides or "rails" are cushioned so that a billiard ball will bounce off and continue rolling. A fundamental skill in the sport of billiards is to put a spin, or "English," on the ball to cause it to bounce off the rail in a desired direction. We will have none of that here.
For the purposes of this problem, we will assume that when the billiard ball bounces off the rail it follows the rule "Angle of Incidence equals Angle of Reflection." In fact, we will refer to such a bounce as a reflection. Additionally, we will assume that our billiard balls are idealized points, and can fit as closely into the corners of our table as desired.
Part A: Show that if a billiard ball reflects consecutively off two adjacent sides of a standard billliard table, then its next reflection must be off one of the other two sides.
Part B: Consider a non-standard billiard table in the shape of an isoceles triangel with angles α, (π-α)/2 and (π-α)/2. After reflecting off the base, what is the largest number of times a billiard ball can reflect between the two legs of the triangle before it must again reflect off the base. The answer may depend on the value of α.