Ellipses+Incircles
I was trying to come up with a rigorous proof of the reflective property of the ellipse. I decided it was a task better left to analysis. However, I did find this little curiousity. You can see the remnants of my endeavour by the point O.
If you want to prove the below, notice that if they really do have the same incircle,. Or try to use the fact that a line like is equal to the sum of its two adjacent sides minus the opposite one divided by 2.