Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute-angled triangle. Let
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
be any line in the plane of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Denote by
![u](/media/m/8/a/3/8a31c0578d8711cccb064f92f92e19ca.png)
,
![v](/media/m/3/d/c/3dc3003f5c10543e81344921fc032374.png)
,
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
the lengths of the perpendiculars to
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
from
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
respectively. Prove the inequality
![u^2\cdot\tan A + v^2\cdot\tan B + w^2\cdot\tan C\geq 2\cdot S](/media/m/8/2/4/8247c204de771be77f6a1903f9e82470.png)
, where
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
is the area of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Determine the lines
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
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.