Let me state two facts:

(1) It is well known that if one takes a point uniformly distributed on the unit circle, and then takes it stereographic projection, the corresponding measure induced on the real line is the standard cauchy distribution.

It is also easy to see that if you take the average of two independent cauchy distributed random variables, it is also cauchy distributed.

My question is the following: Taking cue from (1) above, can we have a geometric proof of (2)?