IMO Shortlist 2007 problem G5
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Avg: 8,0 Let
be a fixed triangle, and let
,
,
be the midpoints of sides
,
,
, respectively. Let
be a variable point on the circumcircle. Let lines
,
,
meet the circumcircle again at
,
,
, respectively. Assume that the points
,
,
,
,
,
are distinct, and lines
,
,
form a triangle. Prove that the area of this triangle does not depend on
.
Author: Christopher Bradley, United Kingdom
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![PA_1](/media/m/f/b/a/fba5e11dce5e2f39c8b6be26e517717c.png)
![PB_1](/media/m/c/4/d/c4de7c19ba6376a5949bb08cd623c087.png)
![PC_1](/media/m/6/5/9/659064fe7ea6e2638a3ed8be830e2054.png)
![A'](/media/m/9/2/6/9267b8bcabe1ad2df0d51dab3364714b.png)
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![A'](/media/m/9/2/6/9267b8bcabe1ad2df0d51dab3364714b.png)
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
![AA'](/media/m/e/1/b/e1bbf498937e07514716e3ab5bd46961.png)
![BB'](/media/m/9/6/c/96c79a8ca3ca0e1f2bc130bf5268932f.png)
![CC'](/media/m/0/0/4/00497a56e1f387a7e764470e33a9447a.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
Author: Christopher Bradley, United Kingdom
Izvor: Međunarodna matematička olimpijada, shortlist 2007