Državno natjecanje 2005 SŠ3 2
Dodao/la:
arhiva1. travnja 2012. Upisana kružnica trokuta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
dodiruje stranice
![\overline{AC}](/media/m/d/9/5/d95354f0f833a5fda9c16a01a878c14f.png)
,
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
i
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
redom u točkama
![M, N](/media/m/7/1/1/7119fd5b5b7ecea73d6b1400ec6abdb7.png)
i
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
. Neka je
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
točka na manjem od dva luka
![MN](/media/m/2/6/7/267a73297a5de9e529d41774ee6ff45a.png)
i
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
tangenta na taj luk s diralištem
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. Tangenta
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
siječe
![\overline{NC}](/media/m/5/c/0/5c072ca7bd5e571f35e44d86a44cf453.png)
i
![\overline{MC}](/media/m/2/9/c/29cdb7f5febf769596ee00cde9230c8d.png)
redom u točkama
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
i
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
. Dokažite da se pravci
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
,
![BQ](/media/m/2/8/c/28cc5d89f53243e9e0fb41492df4736b.png)
,
![SR](/media/m/e/f/d/efd1a1714e4f4f1a33e8d93b070cf783.png)
i
![MN](/media/m/2/6/7/267a73297a5de9e529d41774ee6ff45a.png)
sijeku u jednoj točki.
%V0
Upisana kružnica trokuta $ABC$ dodiruje stranice $\overline{AC}$, $\overline{BC}$ i $\overline{AB}$ redom u točkama $M, N$ i $R$. Neka je $S$ točka na manjem od dva luka $MN$ i $t$ tangenta na taj luk s diralištem $S$. Tangenta $t$ siječe $\overline{NC}$ i $\overline{MC}$ redom u točkama $P$ i $Q$. Dokažite da se pravci $AP$, $BQ$, $SR$ i $MN$ sijeku u jednoj točki.
Izvor: Državno natjecanje iz matematike 2005