IMO Shortlist 2002 problem G1
Kvaliteta:
Avg: 2,0Težina:
Avg: 6,0 Let
be a point on a circle
, and let
be a point distinct from
on the tangent at
to
. Let
be a point not on
such that the line segment
meets
at two distinct points. Let
be the circle touching
at
and touching
at a point
on the opposite side of
from
. Prove that the circumcentre of triangle
lies on the circumcircle of triangle
.
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
![S_2](/media/m/c/1/1/c11855875777bfedb764b27ccc108413.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2002