I found a conjectures of generalization of the Lester circle theorem a more than year ago, but I no a found solution for the conjectures. I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a proof?
Lester theorem: Let $ABC$ be a triangle, then the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.
A generalization of Lester theorem: Let $P$ be a point on the Neuberg cubic. Let $P_A$ be the reflection of $P$ in line $BC$, and define $P_B$ and $P_C$ cyclically. It is known that the lines $AP_A$, $BP_B$, $CP_C$ concur. Let $Q(P)$ be the point of concurrence. Then two Fermat points, $P$, $Q(P)$ lie on a circle.
When $P=X(3)$ (X(3)=the circumcenter of the triangle), it is well-know that $Q(P)=Q(X(3))=X(5)$ (X(5)= the Nine point center of the triangle), the conjucture becomes Lester theorem. (See 1, 2, 3)
1-http://faculty.evansville.edu/ck6/encyclopedia/ETCPart5.html#X7668
2-https://groups.yahoo.com/neo/groups/AdvancedPlaneGeometry/conversations/topics/2546