<Bouncing Barney>

Barney is in the triangular room shown here. He walks from a point on BC parallel to AC. When he reaches AB, he turns and walks parallel to BC. When he reaches AC, he turns and walks parallel to 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? Explore and discuss for various starting points on line BC, including points exterior to segment BC. Discuss and prove any mathematical conjectures you find in the situation.

First, construct Bouncing Barney using GSP.

<step 0> Barney is on the side BC of a given triangle ABC.

<step 1>

Construct the parallel line to AC passing though the point "Barney."

Construct the intersection E of the parallel line and the side AB.

E is the first bouncing point.

<step 2>

Construct the parallel line to BC passing through the point E.

Construct the intersection F of the parallel line and the side AC.

F is the second bouncing point.

<step 3>

Construct the parallel line to AB passing through the point F.

Construct the intersection G of the parallel line and the side BC.

G is the third bouncing point.

<step 4>

Construct the parallel line to AC passing through the point G.

Construct the intersection H of the parallel line and the side AB.

H is the fourth bouncing point.

<step 5>

Construct the parallel line to BC passing through the point H.

Construct the intersection I of the parallel line and the side AC.

I is the fifth bouncing point.

<step 6>

Construct the parallel line to AB passing through the point I.

Then, the intersection of the parallel line and the side BC becomes the starting point.

Therefore, Barney reaches a wall 5 times.

However, what happens if the starting point change? Explore this using the following file.

Drag the point "Barney" on the side BC. 

If you want GSP file, Click HERE!

As you can see, generally Barney hits walls 5 times before returning to the starting point. However, when Barney starts at the midpoint of the side BC, Barney reaches walls just twice.

Now, I will consider arbitrary starting point. Is it possible to happen the same result even if the starting point is not on the side BC?

First, consider the case where the point "Barney" is inside a given triangle ABC.

The process of the construction is the same as the case where "Barney" is on the side BC.

Explore this case using the following file. Drag the point "Barney" inside the triangle ABC and change the shape of the triangle ABC.

If you want GSP file, Click HERE!

As you can explore, Barney hits walls 6 times before returning to the starting point.

Finally, consider the case where the starting point is outside a given triangle ABC.

Explore this case using the following file. Drag the point "Barney" outside the triangle ABC and change the shape of the triangle ABC.

If you want GSP file, Click HERE!

As you can see, the result is similar to the previous case.

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