Circles
and
intersect at points
and
. Distinct points
and
(not at
or
) are selected on
. The lines
and
meet
again at
and
respectively, and the lines
and
meet at
. Prove that, as
and
vary, the circumcentres of triangles
all lie on one fixed circle.
%V0
Circles $S_1$ and $S_2$ intersect at points $P$ and $Q$. Distinct points $A_1$ and $B_1$ (not at $P$ or $Q$) are selected on $S_1$. The lines $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$. Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.
Školjka