IMO Shortlist 2003 problem G2
Kvaliteta:
Avg: 2,5Težina:
Avg: 6,0 Given three fixed pairwisely distinct points
,
,
lying on one straight line in this order. Let
be a circle passing through
and
whose center does not lie on the line
. The tangents to
at
and
intersect each other at a point
. The segment
meets the circle
at
.
Show that the point of intersection of the angle bisector of the angle
with the line
does not depend on the choice of the circle
.
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![PB](/media/m/7/1/c/71c3da8e488f46729e3d8d507af4af81.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
Show that the point of intersection of the angle bisector of the angle
![AQC](/media/m/d/1/6/d162df8c6cbeed0719e9dbf05954f3b9.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2003