Bouncing Barney

by Kyle Schultz

The Problem

Barney is in the triangular room shown here. He walks from a point on Segment BC parallel to Segment AC. When he reaches Segment AB, he turns and walks parallel to Segment BC. When he reaches Segment BC, he turns and walks parallel to Segment AB. Prove that Barney will eventually return to his starting point. How many times will Barney reach a wall before returning to his starting point?

The Solution

The following figures chart Barney's route from the starting point, S.0, to the bouncing points S.1, S.2, etc. Note that segments containing two "S" points, such as S.1 and S.2, will be labeled as Segment S.1:S.2

It appears that a path from S.5 parallel to Segment AB will pass through S.0. To prove this, assume Segment S.0:S.5 is not parallel to Segment AB.

Proof

Consider the figure below:

From the description of Barney's path, three sets of parallel segments are present.

Segments AC, S.3:S.4, and S.0:S.1

Segments AB and S.2:S.3

Segments BC, S.1:S.2, and S.4:S.5

From these parallel segements, parallelograms are formed. These parallelograms are:

B:S.1:S.2:S.3 (labeled in red)

A:S.2:S.3:S.4 (labeled in green)

From the parallel lines and parallelograms, it can be shown that the yellow triangles in the figure below are congruent.

Angle A is congruent to Angle B:S.1:S.0 because they are corresponding angles formed by two parallel lines and a transversal. Angle A:S.4:S.5 and Angle B are congruent for the same reason. Segments A:S.4 and B:S1 are congruent because they are both congruent to Segment S.2:S.3 (opposite sides of a parallelogram).

Thus, the yellow triangles, Triangle A:S.4:S.5 and Triangle S.1:A:S.0 are congruent by angle-side-angle.

From this congruence, Segment S.4:S.5 is congruent to Segment B:S.0 (CPCTC).

Since these segments are congruent and parallel (Segment B:S.0 is part of Segment BC), Quadrilateral B:S.4:S.5:S.0 is a parallelogram. Thus, Segment S.5:S.0 is parallel to Segment AB and must be part of Barney's path.

Thus Barney will return to his starting point after 5 bounces, or six trips across the interior of the triangle.

An exception to this statement occurs when Barney begins at the midpoint of a side of a triangle. In this case, Barney's route will trace the medial triangle and he will return to his starting point after only two bounces (three trips across the interior of the triangle).

Extension

What if Barney began at a point on Line BC, but not on Segement BC? One such path is depicted in the figure below.

Note that all of the sides of Triangle ABC must be extended in order to chart Barney's path. Can you prove that Barney's next move will take him back to S.0? Click Here to access a Geometer's Sketchpad file containg this sketch.