IMO Shortlist 1988 problem 27
Dodao/la:
arhiva2. travnja 2012. Let
be an acute-angled triangle. Let
be any line in the plane of the triangle
. Denote by
,
,
the lengths of the perpendiculars to
from
,
,
respectively. Prove the inequality
, where
is the area of the triangle
. Determine the lines
for which equality holds.
%V0
Let $ABC$ be an acute-angled triangle. Let $L$ be any line in the plane of the triangle $ABC$. Denote by $u$, $v$, $w$ the lengths of the perpendiculars to $L$ from $A$, $B$, $C$ respectively. Prove the inequality $u^2\cdot\tan A + v^2\cdot\tan B + w^2\cdot\tan C\geq 2\cdot S$, where $S$ is the area of the triangle $ABC$. Determine the lines $L$ for which equality holds.
Izvor: Međunarodna matematička olimpijada, shortlist 1988
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