Bouncing Barney


Teresa Cox



Problem:  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 returns and walks parallel to AB. How many times will Barney reach a wall before returning to his starting point?






Click here for a link to a GSP file with this picture.


Does he return to his original starting point?  It certainly looks like it.  But why?


Investigation:  Using the GSP file and continuing to construct parallel lines from each “bounce point”, I discovered Barney will bounce twice on each triangle leg, then return to his starting point.



Even when I change the size of the triangle, the type of triangle, Barney still bounces twice on each wall and eventually returns to his starting point.  Here is an isosceles triangle.



Here is a right triangle.



Now, let’s move the starting point.  As it nears either vertex, the larger inner triangle made by Barney’s path gets larger and approaches the size of the original triangular room, as seen below.


Another interesting situation is when the starting point is the midpoint of the triangle leg.  Barney will bounce only twice before returning to the starting point.


No matter what kind of triangle I create, Barney will only bounce twice if he starts at the midpoint of the first wall.  The triangle created by the midpoints is called the Medial Triangle.


Another observation is the triangles created by any of the situations shown above seem to be similar to the original triangle. 


Summary:  Given the original question, Barney will bounce twice off of each wall and return to his starting point, unless he starts at the midpoint of the wall.  The triangles created by both situations (midpoint and non-midpoint) look similar to the original triangle.  The inner triangle created by the midpoint situation has a special name:  medial triangle.


In the non-midpoint situation, that diagram looks familiar as well, but I have not been able to find any special name associated with it.  At this time I can not explain why Barney only bounces twice.



Author:  Teresa L.  Cox                   Contact me: