Državno natjecanje 2013 SŠ4 4
Dodao/la:
arhiva12. srpnja 2013. Neka su
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
i
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
kružnice s promjerima
![\overline{AP}](/media/m/6/b/a/6ba025fd7238b82fe483707d613e3026.png)
i
![\overline{AQ}](/media/m/0/b/0/0b023f4d7d515e293ddc7cfe2459c85c.png)
. Neka je
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
drugo sjecište kružnica
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
i
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
. Neka je
![Q'](/media/m/6/b/4/6b421d9ca64fe5b597e1c52f0d98e3f0.png)
drugo sjecište kružnice
![k_1](/media/m/3/5/6/35656cbf3adb55dddd30996fc068363b.png)
i pravca
![AQ](/media/m/3/6/7/36700db67d5294dc56876df0725f079d.png)
, a
![P'](/media/m/d/a/0/da0dad9da7f83f64ba8db69b5c83c378.png)
drugo sjecište kružnice
![k_2](/media/m/6/a/b/6abbe24dbf6713b55498fe55ab050d06.png)
i pravca
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
. Kružnica
![k_3](/media/m/a/c/c/acc46117c83b2219c41df2ebf7562d05.png)
prolazi točkama
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
,
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
i
![P'](/media/m/d/a/0/da0dad9da7f83f64ba8db69b5c83c378.png)
, a kružnica
![k_4](/media/m/7/5/8/758e0ae55b105d2a0ada31e1b504de5d.png)
točkama
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
,
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
i
![Q'](/media/m/6/b/4/6b421d9ca64fe5b597e1c52f0d98e3f0.png)
.
Dokaži da pravac na kojem leži zajednička tetiva kružnica
![k_3](/media/m/a/c/c/acc46117c83b2219c41df2ebf7562d05.png)
i
![k_4](/media/m/7/5/8/758e0ae55b105d2a0ada31e1b504de5d.png)
prolazi točkom
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
.
%V0
Neka su $k_1$ i $k_2$ kružnice s promjerima $\overline{AP}$ i $\overline{AQ}$. Neka je $T$ drugo sjecište kružnica $k_1$ i $k_2$. Neka je $Q'$ drugo sjecište kružnice $k_1$ i pravca $AQ$, a $P'$ drugo sjecište kružnice $k_2$ i pravca $AP$. Kružnica $k_3$ prolazi točkama $T$, $P$ i $P'$, a kružnica $k_4$ točkama $T$, $Q$ i $Q'$.
Dokaži da pravac na kojem leži zajednička tetiva kružnica $k_3$ i $k_4$ prolazi točkom $A$.
Izvor: Državno natjecanje iz matematike 2013