Geometry problem!

Question

In the equilateral triangle ABC we send a light beam through The vertex A so that it hits D on side BC so that: BD/BC=1/5. How many times does the light beam have to reflect so that it goes back through vertex A? So the answer for this is 7. And here’s how: Now the question is what happens if instead of BD/BC=1/5 we have BD/BC=3/7?

Answer 1

It won't get into vertex A, it will get to vertex B. Do your problem has a rule how beams reflect from angles? Does it count as one reflection or two?

Answer 2

Try this: instead of reflecting the beam off the side of the triangle, reflect the triangle across the edge hit and let the beam pass through the edge. You’ll get a ray in a triangular grid. Think about how and why this works!