IMO Shortlist 2005 problem G2
Dodao/la:
arhiva2. travnja 2012. Six points are chosen on the sides of an equilateral triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
:
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
,
![A_2](/media/m/a/2/5/a25c6dade4a684fc874981a7d65625f5.png)
on
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
,
![B_2](/media/m/1/8/1/181de00f42000a442a347ff370e521f1.png)
on
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
and
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
,
![C_2](/media/m/a/b/8/ab898e857261e1c35339f3f3d8362ba0.png)
on
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, such that they are the vertices of a convex hexagon
![A_1A_2B_1B_2C_1C_2](/media/m/6/6/0/660d6cc2a88e253b514bf79a96e9ea0f.png)
with equal side lengths.
Prove that the lines
![A_1B_2](/media/m/a/4/3/a43c24b6f56432af6fb0be4bb207e586.png)
,
![B_1C_2](/media/m/b/f/d/bfdb85f6fad20d0e1fcecb83f02e7ee5.png)
and
![C_1A_2](/media/m/0/f/d/0fd2acb2f2ddb3899e19d21060436aa8.png)
are concurrent.
Bogdan Enescu, Romania
%V0
Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.
Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.
Bogdan Enescu, Romania
Izvor: Međunarodna matematička olimpijada, shortlist 2005