1832. Inverted Triangles and Circumcenters On Euler Line

 Let ABC be a triangle and A1B1C1 is perspective with ABC at a point P and homothetic to ABC.

A2 is inversion of A1 respect to circumcircle of ABC. Define B2, C2 cyclically.

* Circumcenter of A2B2C2 lies on the line OP.

Application. Let P be a point on Euler line of ABC. A1,B1,C2 midpoints of AP, BP, CP resp.

A2 is inversion of A1 respect to circumcircle of ABC. Define B2, C2 cyclically.

** O',Circumcenter of A2B2C2 lies on the Euler Line of ABC.

For P=X(2), O'=X(33532)

For P=X(5), O'=X(?)

For P=X(20), O'=X(12084)