A circle
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
with center
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
on base
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
of an isosceles triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is tangent to the equal sides
![AB,AC](/media/m/b/d/4/bd41cfb719b9503047737de42ab652f6.png)
. If point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
on
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and point
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
on
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
are selected such that
![PB \times CQ = (\frac{BC}{2})^2](/media/m/5/3/d/53d8b06cc4668b5772f31987f3f30dcf.png)
, prove that line segment
![PQ](/media/m/f/2/f/f2f65ec376294df7eca22d2c1a189747.png)
is tangent to circle
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
, and prove the converse.
%V0
A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.