Points
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
lie on side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
of acute-angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
so that
![\angle PAB = \angle BCA](/media/m/8/f/b/8fb5f8300ece8473966151823a57da7c.png)
and
![\angle CAQ = \angle ABC](/media/m/e/7/e/e7ea1524f8d8747bee51e08e33127cdd.png)
. Points
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
lie on lines
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
and
![AQ](/media/m/3/6/7/36700db67d5294dc56876df0725f079d.png)
, respectively, such that
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
is the midpoint of
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
, and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
is the midpoint of
![AN](/media/m/6/3/f/63fe98a3ad08df7cdbd7e404dd1aa816.png)
. Prove that lines
![BM](/media/m/9/b/a/9ba306de3378f6c32d1ba470951ff4a4.png)
and
![CN](/media/m/f/1/b/f1bd10dc9b156653548721bc0bf0117a.png)
intersect on the circumcircle of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
%V0
Points $P$ and $Q$ lie on side $BC$ of acute-angled triangle $ABC$ so that $\angle PAB = \angle BCA$ and $\angle CAQ = \angle ABC$. Points $M$ and $N$ lie on lines $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$, and $Q$ is the midpoint of $AN$. Prove that lines $BM$ and $CN$ intersect on the circumcircle of triangle $ABC$.