Bouncing Barney
By
Victor
L. BrunaudVega
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? 












