IMO Shortlist 1998 problem G3
Dodao/la:
arhiva2. travnja 2012. Let
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
be the incenter of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Let
![K,L](/media/m/c/f/8/cf8b7b8c56970a06671ff82ddb7f6450.png)
and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the points of tangency of the incircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
with
![AB,BC](/media/m/3/b/5/3b536b63cbacf75f4074e0c921660bf2.png)
and
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
, respectively. The line
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
passes through
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
and is parallel to
![KL](/media/m/b/1/a/b1ab64b407588444cd365224f9b482b4.png)
. The lines
![MK](/media/m/5/b/e/5bec85a3fcbcd7885b11b39231145af0.png)
and
![ML](/media/m/d/4/a/d4a9f65aad97519d7db147354a9114ef.png)
intersect
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
at the points
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
and
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. Prove that
![\angle RIS](/media/m/9/d/0/9d0088e2086d576b2a3cb6a8413791fd.png)
is acute.
%V0
Let $I$ be the incenter of triangle $ABC$. Let $K,L$ and $M$ be the points of tangency of the incircle of $ABC$ with $AB,BC$ and $CA$, respectively. The line $t$ passes through $B$ and is parallel to $KL$. The lines $MK$ and $ML$ intersect $t$ at the points $R$ and $S$. Prove that $\angle RIS$ is acute.
Izvor: Međunarodna matematička olimpijada, shortlist 1998