Bouncing Barney

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

My construction is as follows and shows that by following the various colored parallel lines, it took Barney 6 steps to return to his starting point.

Click here to see my GSP construction.

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