Conway's Circle: What is it?
A Mathematical Droodle

Motivation

Explanation

The applet helps conceive the right idea: as x changes, the 6 points lie on concentric circles the most remarkable of which is the incircle of the triangle. The incenter of the triangle lies at the intersection of its angle bisectors. Therefore the distances from the incenter to the 6 points are all equal (use the applet with the Hint box checked.)

(Check also the article Around the Incircle for the relation between Conway's circles, Howard Eves's problem and shapes of constant width.)