IMO Shortlist 2001 problem G6
Dodao/la:
arhiva2. travnja 2012. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle and
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
an exterior point in the plane of the triangle. Suppose the lines
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
,
![BP](/media/m/e/e/f/eefb4fe46ab8d85b7067c29b24aa4cfc.png)
,
![CP](/media/m/6/3/0/630424587cadeb75669118dab3df6b98.png)
meet the sides
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
(or extensions thereof) in
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
, respectively. Suppose further that the areas of triangles
![PBD](/media/m/4/b/f/4bfeda912efdb1822b313bb67ee45aa2.png)
,
![PCE](/media/m/7/e/a/7ea686c13e0a0da01a6f8c3482db41cf.png)
,
![PAF](/media/m/6/9/8/698e2eaa6ef7970da6303c4253deb4b5.png)
are all equal. Prove that each of these areas is equal to the area of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
itself.
%V0
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
Izvor: Međunarodna matematička olimpijada, shortlist 2001