Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle.
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
is a point on the side
![(BC)](/media/m/0/2/0/02086af1e260ee528545f340a104dae8.png)
. The line
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
meets the circumcircle again at
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
.
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
is the foot of the perpendicular from
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
is the foot of the perpendicular from
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
to
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
. Show that the line
![PQ](/media/m/f/2/f/f2f65ec376294df7eca22d2c1a189747.png)
is a tangent to the circle on diameter
![XD](/media/m/c/8/7/c87d3f7c358453757aa6e089027ac299.png)
if and only if
![AB = AC](/media/m/9/d/3/9d3b4c44cee3883e9b08931f9d15c670.png)
.
%V0
Let $ABC$ be a triangle. $D$ is a point on the side $(BC)$. The line $AD$ meets the circumcircle again at $X$. $P$ is the foot of the perpendicular from $X$ to $AB$, and $Q$ is the foot of the perpendicular from $X$ to $AC$. Show that the line $PQ$ is a tangent to the circle on diameter $XD$ if and only if $AB = AC$.