IMO Shortlist 1984 problem 15
Dodao/la:
arhiva2. travnja 2012. Angles of a given triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
are all smaller than
![120^\circ](/media/m/f/0/6/f063a54cf240e6d1e674d56b8c9a47a0.png)
. Equilateral triangles
![AFB, BDC](/media/m/f/5/9/f59de72fe6c94ba3fee9e5bf98ccc6d2.png)
and
![CEA](/media/m/c/3/e/c3e7c470abade32bc14b13620818a453.png)
are constructed in the exterior of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
(a) Prove that the lines
![AD, BE](/media/m/9/b/f/9bf7cdc5b45b9c7332abe46f6a8bc880.png)
, and
![CF](/media/m/6/7/0/670c216bc8a05762a60542376587c5fc.png)
pass through one point
![S.](/media/m/3/7/7/3772accbdc4fffed2efa17d53f141907.png)
(b) Prove that
%V0
Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$.
(a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$
(b) Prove that $SD + SE + SF = 2(SA + SB + SC).$
Izvor: Međunarodna matematička olimpijada, shortlist 1984