Bouncing Barney


Victor L. Brunaud-Vega



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.

a)            Prove that Barney will eventually return to his starting point.

b)            How many times will Barney reach a wall before returning to his starting point?

c)            Explore and discuss for various starting points on line BC, including points exterior to segment BC.

d)            Discuss and prove any mathematical conjectures you find in the situation.

 If Barney start to walk from a point of the segment BC, the perimeter of the path followed is equal to the perimeter of triangle ABC.

 You can see a dynamic sketch here. Observe how the perimeters remains equal even if the triangle abc changes its dimensions or the starting point changes its location.

  As Barney always moves following the same angles of the room, he will come to the starting point at the end of his track!



And… what happens if Barney starts to walk out of the room, but following the pattern?


If Barney starts to walk from a point D out of the room but still going in a parallel path to the walls and bouncing when reach the projection of them, the perimeter of the trajectory DEFGHI gets different –bigger- to the perimeter of the triangle ABC.

You can try a dynamic sketch to see this effect here.

Observe that, as Barney still walks following the direction of the walls, the triangles EFC, BGH and ADI are similar to triangle ABC because all of them have the same angle measurements.  In this case, Barney will finish the track at the starting point because he moves on a path parallel to the sides of the triangle ABC all the time.



Barney bounces five times before to reach the starting point again, in the points E, F, G, H and I.  But it looks like there is one exception: if he starts in the middle point of the segment AB.


In this case, Barney bounces only twice before to reach the starting point.



Another interesting situation comes if Barney start to walk from a point located at one of the two points that divide the wall AB of the room in three equal parts.


On this particular case, Barney bounces five times and he walks over the centroid of the triangular room three times.


Is this cool or what?





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