In an acute triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
be the feet of the perpendiculars from the points
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
to the lines
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, respectively, and let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
,
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
,
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
be the feet of the perpendiculars from the points
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
to the lines
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
,
![FD](/media/m/2/b/1/2b1fc9e13464e49208baed61739e3706.png)
,
![DE](/media/m/a/c/d/acdf3f4d3c794d9a897484e9d216f5ec.png)
, respectively.
Prove that
![p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}](/media/m/3/b/a/3ba02af1b46993621fc5cafaa0c9e436.png)
, where
![p\left(T\right)](/media/m/0/2/7/02766d9da212da40c3eb8b8b7cc86fd9.png)
denotes the perimeter of triangle
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
.
%V0
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.
Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .