Let and ,, the symmetrics of vertex over opposite sides. The intersection of the circumcircles of and is . and are defined similarly. Prove that lines , and are concurent.
Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides. The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$. $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.
Školjka 
and 
,
,
the symmetrics of vertex over opposite sides. The intersection of the circumcircles of 
and 
is 
. 
and 
are defined similarly. Prove that lines 
,
and 
are concurent.