Study of Parallel Paths Within

A Triangle

By

# Ricky Joe Hackett

Below is a discussion of the number of times that parallel lines may be constructed from a point on one side of a triangle to the opposite side and parallel to the third side.  First, a description of the problem and the triangle are shown below.

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?

Now that our problem has been defined above, we are ready to create the representation of this problem utilizing Geometer’s Sketch Pad (a product of Key Curriculum Press).

Initially we need to create our triangle by selecting any three points on the sketch pad and then by selecting two points a line segment is created between the points.  Continue this until the triangle has been constructed.  Next the points are labeled to correspond to the above illustration.  Starting at any point on line segment AC and following the directions in the problem, parallel lines are constructed until the image below has been created.

As can be seen from the display, Bouncing Barney does return to his original point of origin.  Next we will attempt to explain why this happens.  The user can view the GSP file if GSP is installed on the user’s computer.