IMO Shortlist 2013 problem G1
Kvaliteta:
Avg: 3,0Težina:
Avg: 6,0 Let
be an acute-angled triangle with orthocenter
, and let
be a point on side
. Denote by
and
the feet of the altitudes from
and
, respectively. Denote by
the circumcircle of
, and let
be the point on
which is diametrically opposite to
. Analogously, denote by
the circumcircle of
, and let
be the point on
which is diametrically opposite to
. Prove that
,
and
are collinear.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
![W](/media/m/2/c/4/2c4aa0f61279d74f16a59bcde17578ef.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![\omega_1](/media/m/2/8/b/28bee4f7488a46b181369fb2b905d1c9.png)
![BWN](/media/m/2/0/4/204706f15f515e64554d56b3501756bf.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![\omega_1](/media/m/2/8/b/28bee4f7488a46b181369fb2b905d1c9.png)
![W](/media/m/2/c/4/2c4aa0f61279d74f16a59bcde17578ef.png)
![\omega_2](/media/m/9/a/7/9a791d5fa32ddb4f4449e9d2d16c7b04.png)
![CWM](/media/m/b/0/c/b0c29c626056c8b8055d0d1f5f3b6f26.png)
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
![\omega_2](/media/m/9/a/7/9a791d5fa32ddb4f4449e9d2d16c7b04.png)
![W](/media/m/2/c/4/2c4aa0f61279d74f16a59bcde17578ef.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
Izvor: Thailand