What happens to the center of the inscribed circle if one vertex of the triangle is moved? Of course, the answer depends on the way it is moved. Interestingly, there is a very nice answer if we move the corner on the circumscribed circle.

 

In this construction, I created a track of the center, while C moves. It seems, the center of the inscribed circles moves on circular arcs.

Click on the animation to stop it.

Can you prove that the two arcs of the track are actually circular?

(Need a hint? Use the chord angle theorem!)

(Need another hint? Computel AMB! It only depends on the angle ACB, which is constant.)

Proof