TITLE: Prove that $EF$ perpendicular to $OI_a$
QUESTION [10 upvotes]: Let $O, I$ and $I_a,$ denote the circumcenter,incenter and excenter in the angle $A$ of a triangle $ABC$. $BI$ meets $AC$ at $E$. $CI$ meets $AB$ at $F$. Prove that $EF$ perpendicular to $OI_a$

It is somewhat like this problem:
Let $H$ be the orthocenter of a triangle $ABC$, and $X, Y, Z$ be the feet of the altitudes from $A, B, C$. $XZ$ meets $HB$ at $E$. $XY$ meets $HC$ at $F$. $O_E$ is center of euler circle of triangle $ABC$. Prove that $AO_E$ perpendicular to $EF$

REPLY [2 votes]: the polar axis of $\triangle II_bI_c$ is $EF$, $OI_a$ is the Eulur line.
Recall that the polar axis of a triangle  perpendicular to the Eulur line, Clearly, $EF\perp OI_a$