IMO Shortlist 1997 problem 9
Dodao/la:
arhiva2. travnja 2012. Let
![A_1A_2A_3](/media/m/b/b/c/bbcede562021e40de971618cb504b791.png)
be a non-isosceles triangle with incenter
![I.](/media/m/6/9/6/696f29d0eea59f984b6c182db592fcf3.png)
Let
![i = 1, 2, 3,](/media/m/8/f/2/8f239e5e13ea63538adb8fc45cd74a32.png)
be the smaller circle through
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
tangent to
![A_iA_{i+1}](/media/m/8/1/7/8177b14dd5ed7502c7574dad25dee836.png)
and
![A_iA_{i+2}](/media/m/4/5/e/45ea0963800cc40d9cd9c9674444ee6c.png)
(the addition of indices being mod 3). Let
![B_i, i = 1, 2, 3,](/media/m/2/e/7/2e70d162b18f65ce043c4eee5373f46a.png)
be the second point of intersection of
![C_{i+1}](/media/m/5/8/a/58aaed598151cfa576260cd98ab773e4.png)
and
![C_{i+2}.](/media/m/4/3/b/43b1f1f321c07216c32654f167ce6319.png)
Prove that the circumcentres of the triangles
![A_1 B_1I,A_2B_2I,A_3B_3I](/media/m/6/3/8/6384e534b25781c027613296166c5cc2.png)
are collinear.
%V0
Let $A_1A_2A_3$ be a non-isosceles triangle with incenter $I.$ Let $C_i,$ $i = 1, 2, 3,$ be the smaller circle through $I$ tangent to $A_iA_{i+1}$ and $A_iA_{i+2}$ (the addition of indices being mod 3). Let $B_i, i = 1, 2, 3,$ be the second point of intersection of $C_{i+1}$ and $C_{i+2}.$ Prove that the circumcentres of the triangles $A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.
Izvor: Međunarodna matematička olimpijada, shortlist 1997