Općinsko natjecanje 2006 SŠ3 3
Dodao/la:
arhiva2. travnja 2012. Na stranici
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
odabrana je bilo koja točka
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
, a na stranici
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
točka
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
, tako da je
![DE](/media/m/a/c/d/acdf3f4d3c794d9a897484e9d216f5ec.png)
paralelno s
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
. Neka su
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
,
![P_1](/media/m/a/8/8/a886eaf7832af6b6b5f56f0ec9a97490.png)
i
![P_2](/media/m/e/c/8/ec8662164615835e6c2307d72a487ec8.png)
redom površine trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
,
![EBD](/media/m/2/3/6/236da002652385b06e35f42d1188a118.png)
i
![ABD](/media/m/a/5/4/a548bc577543629d304ecba1a042f910.png)
, dokaži da je tada
![P_2=\sqrt{PP_1}](/media/m/3/9/6/396bf3cc505103b2e4a47ce0ffa1bc60.png)
.
%V0
Na stranici $\overline{BC}$ trokuta $ABC$ odabrana je bilo koja točka $D$, a na stranici $\overline{AB}$ točka $E$, tako da je $DE$ paralelno s $CA$. Neka su $P$, $P_1$ i $P_2$ redom površine trokuta $ABC$, $EBD$ i $ABD$, dokaži da je tada $P_2=\sqrt{PP_1}$.
Izvor: Općinsko natjecanje iz matematike 2006