For my research I am interested in the transmission characteristics between a transmitter (Tx) and a receiver (Rx) situated in a circular room. In particular, it is important for me to know the number of paths a ray can take such that it reflects *exactly once* off the walls of the room.

Since reflections occur such that the incident ray has the same angle relative to the normal as the reflected ray, I tried to use vectors to attack the problem but the math became very unwieldy.

Empirically, I have found that depending on the situation of the transmitter and receiver, there could be 2, 3, or 4 paths—no more, no less. There is an exceptional case where the transmitter and receiver are co-located at the centre, in which case there are infinitely many paths.

Can my experimental result be validated (or denied) analytically?