Here's the set-up:
Take three circles entered at $(-1,0),(0,0),(1,0)$ with radii $\sqrt 2$ and $1$.
Then pick $p$, any point on the right-hand circle. Reflect $p$ in the horizontal axis to get $-p$. Draw the line segment from $(0,0)$ to $-p$.
Then $q$ is the intersection of this segment with the left circle.
Prove that the distance between $p$ and $q$ is $2$.
I have proved this using complex analysis, but is there an intuitive way to do this using pure geometry?