A point is chosen inside a triangle . Let , , and be the reflections of in , , and , respectively. Let be the circumcircle of the triangle . The lines , , and meet again at , , and , respectively. Prove that the lines , , and are concurrent on .
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
Školjka 
is chosen inside a triangle 
. Let 
, 
, and 
be the reflections of 
, 
, and 
, respectively. Let 
be the circumcircle of the triangle 
. The lines 
, 
, and 
meet 
, 
, and 
, respectively. Prove that the lines 
, 
, and 
are concurrent on