Državno natjecanje 2001 SŠ2 2
Dodao/la:
arhiva1. travnja 2012. Kružnica sa središtem
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
dira stranicu
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
i produžetke stranica
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
i
![\overline{AC}](/media/m/d/9/5/d95354f0f833a5fda9c16a01a878c14f.png)
trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
redom u točkama
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
,
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
i
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
. Dužine
![\overline{OB}](/media/m/1/9/4/194d1eb9451d850c84fe696dcdf025d2.png)
i
![\overline{OC}](/media/m/f/9/6/f96f38446d82bd9c3625dc03dfafbf74.png)
sijeku spojnicu
![\overline{PQ}](/media/m/f/5/d/f5d8cee1cf25e485433f6a207a20ad45.png)
redom u točkama
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
i
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
. Dokažite da je
%V0
Kružnica sa središtem $O$ dira stranicu $\overline{BC}$ i produžetke stranica $\overline{AB}$ i $\overline{AC}$ trokuta $ABC$ redom u točkama $K$, $P$ i $Q$. Dužine $\overline{OB}$ i $\overline{OC}$ sijeku spojnicu $\overline{PQ}$ redom u točkama $M$ i $N$. Dokažite da je $$\frac{|QN|}{|AB|} = \frac{|MN|}{|BC|} = \frac{|MP|}{|CA|} \text{.}$$
Izvor: Državno natjecanje iz matematike 2001