Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle with
![\angle B > \angle C](/media/m/f/b/c/fbcf95fc3cf2504af0093e150fe7de36.png)
. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
be two different points on line
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
such that
![\angle PBA = \angle QBA = \angle ACB](/media/m/9/7/a/97ae01a0a74fe157acd4b73e6a225f4a.png)
and
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
is located between
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
. Suppose that there exists and interior point
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
of segment
![BQ](/media/m/2/8/c/28cc5d89f53243e9e0fb41492df4736b.png)
for which
![PD = PB](/media/m/1/e/d/1ed9c487682fd68da7fe03ed26ebc8ea.png)
. Let the ray
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
intersect the circle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
at
![R \neq A](/media/m/3/b/c/3bc3b52bb660b2c52bab72534f53d8a8.png)
. Prove that
![QB = QR](/media/m/b/e/3/be3845b22dc492ac673e5cc31c63abc0.png)
.
%V0
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB$ and $A$ is located between $P$ and $C$. Suppose that there exists and interior point $D$ of segment $BQ$ for which $PD = PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.