Županijsko natjecanje 2003 SŠ3 3
Dodao/la:
arhiva1. travnja 2012. Na strani
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
trostrane piramide
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
dana je točka
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
, kroz koju su povučene dužine
![\overline{OA_1}](/media/m/1/6/c/16c75e52a72dfe85bbd37f640771e6be.png)
,
![\overline{OB_1}](/media/m/3/6/3/3631aaf8fc5f8c7c8732cff876e4804e.png)
i
![\overline{OC_1}](/media/m/a/f/8/af8ea456b198f1d9b7b8f96c80444ae2.png)
, paralelno s bridovima
![\overline{DA}](/media/m/8/4/5/845d2fffb3c5eb6412b26a001c3b4b4d.png)
,
![\overline{DB}](/media/m/5/4/c/54ce0e9ace40478bd6fd1b4235465f01.png)
i
![\overline{DC}](/media/m/d/4/8/d4833b07ff9ba6f723009f06316626fd.png)
, do presjeka
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
,
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
,
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
sa stranama piramide. Dokažite da je
%V0
Na strani $ABC$ trostrane piramide $ABCD$ dana je točka $O$, kroz koju su povučene dužine $\overline{OA_1}$, $\overline{OB_1}$ i $\overline{OC_1}$, paralelno s bridovima $\overline{DA}$, $\overline{DB}$ i $\overline{DC}$, do presjeka $A_1$, $B_1$, $C_1$ sa stranama piramide. Dokažite da je $$
\dfrac{|OA_1|}{|DA|}+\dfrac{|OB_1|}{|DB|}+\dfrac{|OC_1|}{|DC|}=1.
$$
Izvor: Županijsko natjecanje iz matematike 2003