Neka su
![h_1](/media/m/4/0/5/405fe34684b6152379543cc1eecd9f00.png)
,
![h_2](/media/m/5/d/d/5dd34799df33b7962251de451023a5a5.png)
,
![h_3](/media/m/2/d/3/2d322048c3839657a6b6f35329bfb154.png)
duljine visina trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
na stranice
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
,
![\overline{CA}](/media/m/c/e/9/ce9fb8497710464615e1d00d148c5663.png)
,
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
, redom, a
![u](/media/m/8/a/3/8a31c0578d8711cccb064f92f92e19ca.png)
,
![v](/media/m/3/d/c/3dc3003f5c10543e81344921fc032374.png)
,
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
udaljenosti točke
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
iz unutrašnjosti trokuta od stranica
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
,
![\overline{CA}](/media/m/c/e/9/ce9fb8497710464615e1d00d148c5663.png)
,
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
. Dokažite:
![\frac{h_1}{u} + \frac{h_2}{v} + \frac{h_3}{w} \geq 9,](/media/m/a/6/d/a6d3315ee370d982aae504ffe568d5f1.png)
![h_1h_2h_3 \geq 27uvw,](/media/m/8/3/c/83cf402bb1aa9289255b84c3573b7307.png)
%V0
Neka su $h_1$, $h_2$, $h_3$ duljine visina trokuta $ABC$ na stranice $\overline{BC}$, $\overline{CA}$, $\overline{AB}$, redom, a $u$, $v$, $w$ udaljenosti točke $M$ iz unutrašnjosti trokuta od stranica $\overline{BC}$, $\overline{CA}$, $\overline{AB}$. Dokažite:
$$\frac{h_1}{u} + \frac{h_2}{v} + \frac{h_3}{w} \geq 9,$$
$$h_1h_2h_3 \geq 27uvw,$$
$$(h_1-u)(h_2-v)(h_3-w) \geq 8uvw.$$