Zadan je trokut
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
s visinama
![h_a](/media/m/f/0/d/f0d769bf3f52b4178084ce3b3d441a05.png)
,
![h_b](/media/m/d/0/6/d06cba1bca5b7f72a98766245f3d8085.png)
,
![h_c](/media/m/b/a/1/ba168021897f698e7b085b67f0963cb8.png)
. Sjecišta simetrala kutova s nasuprotnim stranicama označimo s
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
, a udaljenosti točaka
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
od pravaca
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
,
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
redom sa
![d_a](/media/m/a/7/e/a7eed27e509eb9f1cc3a667ce155cca3.png)
,
![d_b](/media/m/8/0/1/8011ac58ace8f6d331e13011cfe64a27.png)
,
![d_c](/media/m/6/9/c/69cec64b135c69bb1cf8ca66089db993.png)
. Dokažite nejednakost
%V0
Zadan je trokut $ABC$ s visinama $h_a$, $h_b$, $h_c$. Sjecišta simetrala kutova s nasuprotnim stranicama označimo s $D$, $E$, $F$, a udaljenosti točaka $D$, $E$, $F$ od pravaca $AB$, $BC$, $CA$ redom sa $d_a$, $d_b$, $d_c$. Dokažite nejednakost $$\frac{d_a}{h_a} + \frac{d_b}{h_b} + \frac{d_c}{h_c} \geqslant \frac32 \text{.}$$