A saw a neat math gif at mathani on Tumblr
[url]http://mathani.tumblr.com/post/77482183135/two-circles-intersect-at-two-points-as-a-point[/url]
This sketch started because of the "I have to make this" moment, but then I added a 2nd chord in the other circle, and an unconstrained point to compare it with.
So why is the segment of constant length?