Let
![A,B,C,D](/media/m/8/5/d/85d135de173dbb765c7a2f175c5b2f60.png)
be four distinct points on a line, in that order. The circles with diameters
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
intersect at
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
. The line
![XY](/media/m/1/c/e/1ce2b6bc5783d5ee7b3276a845f41d6e.png)
meets
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a point on the line
![XY](/media/m/1/c/e/1ce2b6bc5783d5ee7b3276a845f41d6e.png)
other than
![Z](/media/m/7/9/4/794ff2bd637e30ea27e50e57eecd0b76.png)
. The line
![CP](/media/m/6/3/0/630424587cadeb75669118dab3df6b98.png)
intersects the circle with diameter
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
, and the line
![BP](/media/m/e/e/f/eefb4fe46ab8d85b7067c29b24aa4cfc.png)
intersects the circle with diameter
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
at
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
. Prove that the lines
![AM,DN,XY](/media/m/3/6/2/3628ecadd4d8a8bf2ac80a4c661a4382.png)
are concurrent.
%V0
Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM,DN,XY$ are concurrent.