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Many pool players already know this simple mathematical lesson, since it comes up every time you carom the cue ball off a rail. This law tells you that the angle at which the ball strikes the rail is equal to the angle the ball bounces off at. In other words, if the ball approaches the rail at a 30º angle, it will bounce off at a 30º angle as well. The law of reflection originally refers to the behavior of light. It's usually written "the angle of incidence is equal to the angle of reflection." In this scenario, the goal is to carom the cue ball off the rail, and have it return to strike the object ball. Now set up a basic geometry problem as possible:  Imagine a line from the cue ball to the rail, intersecting at right angles. Now imagine the cue ball traveling to the rail. This path is the hypotenuse of a right triangle, formed by your first line and a section of the rail. Now picture the cue ball bouncing off and hitting the object ball. Mentally draw a second right triangle pointing the opposite direction. In this case, we can use the "Angle Angle Side" rule. If both triangles have two equal angles and one equal side (in the same configuration), the two triangles are congruent. (In other words, they are the same shape and size). We can prove that these triangles meet these conditions:  The law of reflection tells us that the two angles between the hypotenuses and the rail are equal. Both are right triangles, so they each have two 90º angles. Since the two balls started equidistant from the rail, we know the two sides between the ball and the rail are equal. Since the two triangles are congruent, the two sides that lie along the rail are also equal to each other. This means the point where the cue ball strikes the rail is equidistant from the two starting positions of the ball. Aim for this midpoint whenever the two balls are an equal distant from the rail. Let's say the cue ball is twice as far from the rail as the object ball. You can still picture two right triangles formed by the cue ball's ideal path, and use intuitive geometry to guide your aim:  The two triangles still share the same angles, but not the same lengths. This makes them similar triangles: same shape, different sizes. Since the cue ball is twice as far from the rail, the first triangle is twice as large as the second triangle. This means the first triangle's "rail side" is twice as long as the second triangle's "rail side." Aim for a point on the rail ⅔ of the way to the object ball, since ⅔ is twice as long as ⅓.
Understand the law of reflection. Set up the cue ball and object ball equidistant from the rail. Prove the two triangles are congruent. Aim at the midpoint of the rail section. Use similar triangles if the balls are not equidistant from the rail.