U trokutu
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
dane su točke
![D, E, F](/media/m/e/c/a/ecac01ab91092791814283e516ec0b5a.png)
na stranicama
![\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)
tako da je
![|AF|=\frac{1}{n}|AB|](/media/m/2/3/a/23abdb70429038eb1fcf66fae145ed57.png)
,
![|BD|=\frac{1}{n}|BC|](/media/m/8/e/4/8e47a5bcd4240c142109691428119807.png)
,
![|CE|=\frac{1}{n}|CA|](/media/m/e/2/3/e23d783aa04738f5fff9999ebc88faad.png)
. Pravci
![AD, BE, CF](/media/m/f/2/a/f2a09dbac9590479b49fd997195c5476.png)
sijeku se u točkama
![G, H, I](/media/m/8/6/5/86531dde45f097f3da583ad80a4e2e83.png)
. Pokažite da je površina trokuta
![GHI](/media/m/7/2/5/725d1740b72722f3c1fd24cce7cc0dbe.png)
jednaka
%V0
U trokutu $ABC$ dane su točke $D, E, F$ na stranicama $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ tako da je $|AF|=\frac{1}{n}|AB|$, $|BD|=\frac{1}{n}|BC|$, $|CE|=\frac{1}{n}|CA|$. Pravci $AD, BE, CF$ sijeku se u točkama $G, H, I$. Pokažite da je površina trokuta $GHI$ jednaka
$$ P_{GHI} = \frac{(n-2)^2}{n^2-n+1} P_{ABC} .$$