IMO Shortlist 2009 problem G6
Kvaliteta:
Avg: 3,0Težina:
Avg: 8,0 Let the sides
and
of the quadrilateral
(such that
is not parallel to
) intersect at point
. Points
and
are circumcenters and points
and
are orthocenters of triangles
and
, respectively. Denote the midpoints of segments
and
by
and
, respectively. Prove that the perpendicular from
on
, the perpendicular from
on
and the lines
are concurrent.
Proposed by Ukraine
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![O_1](/media/m/7/2/b/72b270d556043f6f393afbf50620eb57.png)
![O_2](/media/m/f/2/d/f2de7ab4fb5625160a4d2f2ac2dd707d.png)
![H_1](/media/m/4/3/8/438f902eeccaa8bef218d8a558a31be3.png)
![H_2](/media/m/d/2/7/d27436e2be865ceae1b48b49590f0e8b.png)
![ABP](/media/m/b/3/e/b3e28c035130f58ee161d3f0a9639e17.png)
![CDP](/media/m/c/d/d/cdd9630104b89b454c13729dc4b8ab47.png)
![O_1H_1](/media/m/1/1/b/11be90c73b799c8d049ebdf884a3989a.png)
![O_2H_2](/media/m/1/8/4/184f8215166aae4e6d1e4e5c8a84680b.png)
![E_1](/media/m/4/8/8/488028e5be310a0667dd7afafb1e6a96.png)
![E_2](/media/m/8/f/1/8f1fc86b15e615867fef182490fa4a5a.png)
![E_1](/media/m/4/8/8/488028e5be310a0667dd7afafb1e6a96.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![E_2](/media/m/8/f/1/8f1fc86b15e615867fef182490fa4a5a.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![H_1H_2](/media/m/6/3/2/632ed4fa03466055dfa5d43b090a68ce.png)
Proposed by Ukraine
Izvor: Međunarodna matematička olimpijada, shortlist 2009