 Billiards in a triangle
Would that be trilliards?

by

D Hembree

Imagine a triangular billiards table. This page offers you the chance to investigate two simple questions:

1) Is it possible to shoot a ball and have it return to its starting point?

2) Is it possible to shoot a ball and have it bounce off each side of the triangle and then retrace its path?

Recall from physics that, assuming there is no spin on a ball, it will bounce off a straight object so that the angle of reflection equals the angle of incidence. Or, as a hint for later, the path of a ball as it leaves the object is the reflection of the path of approach across a perpendicular constructed at the point of contact.

You will have a chance to use Geometer's Sketchpad to try to answer these and other questions for billiard tables with acute angles. For simplicity, we will restrict the starting point for the bouncing ball to lie on one side of the triangle as shown below. You may click here or anywhere on the figure to open GSP and drag either the beginning point or the point labeled "aim at this point" to see the path of the ball through the first 6 bounces. Can you succeed with question 1? Recall the task is to locate the beginning point and the target point so that the ball bounces back and hits the beginning point within the first 6 bounces. If you want to see a picture of how I did it click here (use your browser's BACK button to return to this page), but try it yourself first. Notice that my shot returned to the start after the fourth bounce, but it was not in the process of retracing its path, as shown by the dashed lines. There are lots of ways to return to the beginning. Try to do it after the second bounce, the third bounce, the fourth, . . .

Now for the main question: Can you make the ball return to its starting point and then retrace its path?

Can you do it after only one circuit (i.e. after two bounces)?

Click on the figure above and explore. It's worth some time to try and accomplish this, since the locations of the starting point and target point are very specific to the triangle.
If you give up, you can see an image of a solution for the ball to retrace its path here, or an image of the unique solution to retracing its path after one bounce by clicking here.

What's so special about the location of the starting point and the target point if the ball is to retrace its path after one circuit of the triangle? I bet you can discover it if you haven't already. Here's a GSP file like the one above with something added as a hint. Play with it before you read on.

Did it surprise you that the locations for the start and target points are the "feet " of the altitudes of the original triangle? Notice that the altitudes must also bisect the angles of the ball's path. If you show the solution and then drag the start and target points, you'll also discover something about how to make the ball retrace its path after two circuits.

The path of the ball as it retraces its path after one circuit called the pedal triangle of the original triangle. A pedal triangle is formed by connecting the feet of the altitudes of any triangle.

Questions and extensions:
Can you prove that the pedal triangle is the path the ball must take?
Can you continue the constructions in GSP to show more bounces and get a 3 circuit solution?
Are there solutions starting with a right triangle? An obtuse triangle?