**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. Prove that Barney will eventually return
to his starting point. How many times will Barney reach a wall
before returning to his starting point? Explore and discuss for
various starting points on line BC, including points exterior
to segment BC. Discuss and prove any mathematical conjectures
you find in the situation.**

So what we have found from our explorations, is that Barney always returns to his starting point, and that it either takes him three or six paths to get back to his point of origin. If we take a look at our case where Barney starts at one of the vertices, we see that his distanced "bounced"/ traveled is equal to the perimeter of our triangle. We will us GSP to investigate our cases above and compare the perimeter of our triangle to the length of Barney's journey.

Click here to view calculated measurements of Barney's journey compared to our triangle.

We see that if Barney's point of origin is inside our triangle then the length of his journey IS equal to the perimeter of our triangle. We will use the properties of a parrallelogram to prove our findings....

We will first define a paralleogram and point out properties we will use......

**In geometry, a parallelogram
is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram
are of equal length, and the opposite
angles of a parallelogram are congruent.**

Let's take a look at the different parallelograms in our Triangle and then use the fact that opposite sides of a parallelogram are equal.

Well, we see once again he does get back to his starting point, and he takes six paths to get there. But in this case the length of his journey is obviously not equal to the perimeter of our triangle.

When we look at the length of Barney's Journey, we see that it is obviously not equal to the perimeter of the triangle, so we will use directed measurements to investigate further.

__So in summation we have found:__

1) Barney will always return to his original starting point

2) If Barney starts inside the triangle:

A)

starts at midpoint of side: it will take him three paths to return to his origin and the length of his journey will equal 1/2 perimeter of our triangle; or he can go around twice and the length of his journey will be equal to the perimeter of our triangle.B)

starts at vertices of triangle: his journey consists of 3 paths and his journey is equal to the perimeter of our triangle, in fact his journey is the original triangle.C)

if he starts at any other point inside our triangle: he returns to his starting point using six paths and the length of his journey is equal to the perimeter of our triangle.

3) If Barney starts outside the triangle, it takes him six paths to return to his starting point and the length of his journey using directed measurements is equal, once again, to the perimeter of our triangle.

CLICK HERE if you wish to explore Bouncing Barney further.