Let two circles
![S_{1}](/media/m/c/3/2/c32b2ccf37ad57aa0ac979e8922c268a.png)
and
![S_{2}](/media/m/a/9/4/a94de2299932dfc5d3bb6afed0ea0bf3.png)
meet at the points
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
. A line through
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
meets
![S_{1}](/media/m/c/3/2/c32b2ccf37ad57aa0ac979e8922c268a.png)
again at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![S_{2}](/media/m/a/9/4/a94de2299932dfc5d3bb6afed0ea0bf3.png)
again at
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
,
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
,
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
be three points on the line segments
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
,
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
respectively, with
![MN](/media/m/2/6/7/267a73297a5de9e529d41774ee6ff45a.png)
parallel to
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
and
![MK](/media/m/5/b/e/5bec85a3fcbcd7885b11b39231145af0.png)
parallel to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Let
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
be points on those arcs
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
of
![S_{1}](/media/m/c/3/2/c32b2ccf37ad57aa0ac979e8922c268a.png)
and
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
of
![S_{2}](/media/m/a/9/4/a94de2299932dfc5d3bb6afed0ea0bf3.png)
respectively that do not contain
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
. Given that
![EN](/media/m/8/a/7/8a738873fd9bb9b92a40601aa803337f.png)
is perpendicular to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
and
![FK](/media/m/1/8/5/185a554a2095520f3032542c84f6aa69.png)
is perpendicular to
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
prove that
![\angle EMF=90^{\circ}](/media/m/f/f/0/ff0920ba292e7bd9765ea24e3dfe5abe.png)
.
%V0
Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$. A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$. Let $M$, $N$, $K$ be three points on the line segments $CD$, $BC$, $BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$ and $F$ be points on those arcs $BC$ of $S_{1}$ and $BD$ of $S_{2}$ respectively that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF=90^{\circ}$.