IMO Shortlist 2000 problem G1
Dodao/la:
arhiva2. travnja 2012. In the plane we are given two circles intersecting at
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
. Prove that there exist four points with the following property:
(P) For every circle touching the two given circles at
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
, and meeting the line
![XY](/media/m/1/c/e/1ce2b6bc5783d5ee7b3276a845f41d6e.png)
at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
, each of the lines
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
,
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
,
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
passes through one of these points.
%V0
In the plane we are given two circles intersecting at $X$ and $Y$. Prove that there exist four points with the following property:
(P) For every circle touching the two given circles at $A$ and $B$, and meeting the line $XY$ at $C$ and $D$, each of the lines $AC$, $AD$, $BC$, $BD$ passes through one of these points.
Izvor: Međunarodna matematička olimpijada, shortlist 2000