Two circles
![G_1](/media/m/b/c/d/bcd9cf8ee60f35cdf346a1ce12a31bc1.png)
and
![G_2](/media/m/8/a/6/8a68fbfad95710d3eddbccfae32746f0.png)
intersect at two points
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
. Let
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
be the line tangent to these circles at
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
, respectively, so that
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
lies closer to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
than
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
. Let
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
be the line parallel to
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and passing through the point
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
, with
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
on
![G_1](/media/m/b/c/d/bcd9cf8ee60f35cdf346a1ce12a31bc1.png)
and
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
on
![G_2](/media/m/8/a/6/8a68fbfad95710d3eddbccfae32746f0.png)
. Lines
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
meet at
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
; lines
![AN](/media/m/6/3/f/63fe98a3ad08df7cdbd7e404dd1aa816.png)
and
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
meet at
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
; lines
![BN](/media/m/9/2/3/923313310d49dd4405c2a3573960a679.png)
and
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
meet at
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
. Show that
![EP = EQ](/media/m/a/7/9/a794d286ac59fa9c7e5173d8a301876f.png)
.
%V0
Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$. Let $AB$ be the line tangent to these circles at $A$ and $B$, respectively, so that $M$ lies closer to $AB$ than $N$. Let $CD$ be the line parallel to $AB$ and passing through the point $M$, with $C$ on $G_1$ and $D$ on $G_2$. Lines $AC$ and $BD$ meet at $E$; lines $AN$ and $CD$ meet at $P$; lines $BN$ and $CD$ meet at $Q$. Show that $EP = EQ$.