93-Four Jerabek Centers lie on a circle

Let $ABC$ be a triangle, let the Euler line of $ABC$ meets $BC,CA,AB$ at $A_1,B_1,C_1$ respectively.

Define $A_2=X_{125}$ of the triangle $AB_1C_1$. Define $B_2,C_2$ cyclically. 

1-Then circumcenter of the triangle $A_2B_2C_2$ is Gossard perpector

2-Two triangle $A_2B_2C_2$ and ABC are similar and perpective, Which is the perpector?

3-$X(125)$ of ABC also lie on circumcenter of $A_2B_2C_2$ (Four point $X(125)$ lie on a circle)

4-Let $A_3B_3C_3$ be the paralogic triangle of ABC whose perpectrix is Euler line, then  $A_2B_2C_2$ perpective with $A_3B_3C_3$, which is the perpector?

5-Circumcircle of $(A_3B_3C_3), (A_2B_2C_2)$ and $(ABC)$ concurrent at one point, which point?