Here is a problem involving curvilinear incircles and mixtilinear incircles.
Let a triangle$\triangle$$ABC$ have circumcircle $\gamma$.It's A-Excircle tangency point at side$BC$ is $D$ Let $\gamma_1$ be the circle tangent to $AD$,$BD$,$\gamma$ also $\gamma_2$ is the circle tangent to $AD$,$CD$,and$\gamma$.prove that $\gamma_1$ and $\gamma_2$ are congruent.
I have tried to prove the converse of the problem.
I drew two congruent circles on the same side of $BC$,$\gamma_1$ and $\gamma_2$ tangent to $\gamma$ and $BC$.Let the tangency points of $\gamma$ and $\gamma_1$ are $\alpha_1$ and the tangency point of $\gamma_2$ and $\gamma$ be $\alpha_2$.Let the tangents of $\gamma$ at $\alpha_1$ and $\alpha_2$ intersect at point $M$.Invert around the circle centered at $M$ and orthogonal to $\gamma$.Then I got stuck.
I have thought of another approach .according to Thebault's theorem,$P$,$I$,$Q$ are collinear where $P$,$Q$,$I$ are the centers of $\gamma_1$,$\gamma_2$and the incircle of $\triangle$ABC. Then I tried to prove the congruency of the incircle and $\gamma_1$ also The congruency of the incircle and $\gamma_2$.Also note that,$\angle$$BAD$=$\angle$$CAK$.where $K$ is the tangency point of $\gamma$ and the A-mixtilinear circle of $\triangle$$ABC$.
I have tried two-approach but failed to go any further. Can someone help me?
