IMO Shortlist 1995 problem G8
Kvaliteta:
Avg: 0,0Težina:
Avg: 9,0 Suppose that
is a cyclic quadrilateral. Let
and
. Denote by
and
the orthocenters of triangles
and
, respectively. Prove that the points
,
,
are collinear.
Original formulation:
Let
be a triangle. A circle passing through
and
intersects the sides
and
again at
and
respectively. Prove that
,
and
are concurrent, where
and
are the orthocentres of triangles
and
respectively.
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![E = AC\cap BD](/media/m/3/3/0/33093b26f5a801e16c1dad73b735d267.png)
![F = AB\cap CD](/media/m/3/6/0/360f366c801a36de25b18154b206f1ce.png)
![H_{1}](/media/m/0/a/9/0a93c8caa84eae4a39086a6fcaa04623.png)
![H_{2}](/media/m/7/6/4/76441deb0d4a7d4b74722138b41c6769.png)
![EAD](/media/m/c/5/a/c5ac81537aa919a08d09a56bf59ecc80.png)
![EBC](/media/m/9/0/1/901a1afb6a0a1a1ac79ecf8925bd1f8a.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![H_{1}](/media/m/0/a/9/0a93c8caa84eae4a39086a6fcaa04623.png)
![H_{2}](/media/m/7/6/4/76441deb0d4a7d4b74722138b41c6769.png)
Original formulation:
Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
![B',](/media/m/7/1/6/716240d989aa23ce2d90f6d67379283e.png)
![BB'](/media/m/9/6/c/96c79a8ca3ca0e1f2bc130bf5268932f.png)
![CC'](/media/m/0/0/4/00497a56e1f387a7e764470e33a9447a.png)
![HH'](/media/m/0/0/4/0040cfb37ddd22af9fde743ee9258636.png)
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
![H'](/media/m/a/2/4/a2471a30be38de833e7514bf11275db3.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![AB'C'](/media/m/3/8/4/384f3919c233da9fabbbe2aa90b0bd0d.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1995