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A constant of two circles

A saw a neat math gif at mathani on Tumblr http://mathani.tumblr.com/post/77482183135/two-circles-intersect-at-two-points-as-a-point 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?