Circles
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
and
![S_2](/media/m/c/1/1/c11855875777bfedb764b27ccc108413.png)
intersect at points
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
. Distinct points
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
and
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
(not at
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
or
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
) are selected on
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
. The lines
![A_1P](/media/m/3/b/6/3b646e0473aab4fc0736ae8acaa12288.png)
and
![B_1P](/media/m/e/d/3/ed3f252e668bc82a56478c6bc5102bb8.png)
meet
![S_2](/media/m/c/1/1/c11855875777bfedb764b27ccc108413.png)
again at
![A_2](/media/m/a/2/5/a25c6dade4a684fc874981a7d65625f5.png)
and
![B_2](/media/m/1/8/1/181de00f42000a442a347ff370e521f1.png)
respectively, and the lines
![A_1B_1](/media/m/9/f/6/9f6ed35a7c55373dfba47e03dd22bf85.png)
and
![A_2B_2](/media/m/f/e/0/fe0cfcabb76cf8315d86c22e6f8dee9f.png)
meet at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
. Prove that, as
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
and
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
vary, the circumcentres of triangles
![A_1A_2C](/media/m/3/3/d/33ddf72df9888d30db52120b47cb81de.png)
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.