Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim:

Claim.Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in sideline $AB$, and define $E$ and $F$ cyclically. The lines $CD$, $BF$, $AE$ concur in X(79) . Then, the two Fermat points , incenter and $X(79)$ lie on the same circle.

GeoGebra applet that demonstrates this claim can be found here.