Let
![X,Y,Z](/media/m/7/9/c/79c32eeaaac576ea9200a5efaa993302.png)
be the midpoints of the small arcs
![BC,CA,AB](/media/m/9/8/c/98c204ffa459114826231180fce7ec09.png)
respectively (arcs of the circumcircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
).
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is an arbitrary point on
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
, and the parallels through
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
to the internal bisectors of
![\angle B,\angle C](/media/m/1/6/2/162923e6d9b04ace3b4c811cdee7b848.png)
cut the external bisectors of
![\angle C,\angle B](/media/m/f/7/4/f741d3b52485e3bcf834dda4d2f9bdb8.png)
in
![N,P](/media/m/0/b/6/0b6a88f87e5f20718e54f400d7d26f9e.png)
respectively. Show that
![XM,YN,ZP](/media/m/f/1/a/f1aafc0870a87eba6d5bc2e5f3e6c8b3.png)
concur.
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Let $X,Y,Z$ be the midpoints of the small arcs $BC,CA,AB$ respectively (arcs of the circumcircle of $ABC$). $M$ is an arbitrary point on $BC$, and the parallels through $M$ to the internal bisectors of $\angle B,\angle C$ cut the external bisectors of $\angle C,\angle B$ in $N,P$ respectively. Show that $XM,YN,ZP$ concur.