# Distance between particular points on two circles

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?

• What’s the point of having the middle circle?
– amd
Commented Mar 5, 2016 at 21:57
• @amd to confuse both the enemies and allies, but mostly the student.
– CAGT
Commented Mar 5, 2016 at 22:33
• It could be a hint...
– Moti
Commented Mar 5, 2016 at 22:55
• I would just say it intuitively... since the centers of the circles are exactly 2 units away... all corresponding projections of an arc onto another arc, whose arc curvatures are identical, the distance will be the separation of the centers.
– CAGT
Commented Mar 5, 2016 at 23:13