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)?
Feller interpretes (the general version of) your second paragraph as a version of Huygens' principle, see the footnote on page 51 of An Introduction to Probability and Its Applications, Vol. II, Second Edition 1971, with a credit to J.W. Walsh.
Maybe something like this will work.
Consider $U_1$ and $U_2$ drawn uniformly at random on the unit circle. Because they are uniformly distributed, we may rotate the circle until $U_1$ is at the ``north pole" without altering their distributions. Now, stereographic projection takes $U_1$ to $Z_1 = 0$ and $U_2$ to some point $Z_2$. Then $\frac{1}{2}(Z_1 + Z_2) = \frac{Z_2}{2}$, which is Cauchy distributed with scale $\frac{1}{2}$ by your point (1). However, noting that we had two directions we could have rotated -- corresponding to two distinct projections of $U_2$ -- we recover our factor of 2, giving the result (2).
