Državno natjecanje 2002 SŠ2 3
Dodao/la:
arhiva1. travnja 2012. Ako za trokute s duljinama stranica
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
,
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
i
![a^\prime](/media/m/8/c/e/8cedd22aae12170388667f3a5e9b0b1e.png)
,
![b^\prime](/media/m/9/9/2/99202b998d21e444db0e5312fdc8376e.png)
,
![c^\prime](/media/m/b/9/7/b97a6a245dbeb22c8bf030d8f383fedd.png)
te nasuprotnim kutovima
![\alpha](/media/m/f/c/3/fc35d340e96ae7906bf381cae06e4d59.png)
,
![\beta](/media/m/c/e/f/cef1e3bcf491ef3475085d09fd7d291e.png)
,
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
i
![\alpha^\prime](/media/m/0/7/4/0745af99713fa5f71d61e353505b35bb.png)
,
![\beta^\prime](/media/m/f/3/2/f3256d45b328651f4dbd8588d67fcd04.png)
,
![\gamma^\prime](/media/m/d/2/1/d2178e793274e2d52c07b11480df14ff.png)
vrijede jednakosti
![\alpha + \alpha^\prime = \pi](/media/m/0/d/4/0d4d291d67a69821744a2051f5477598.png)
i
![\beta = \beta^\prime](/media/m/1/c/4/1c4ccfc47baa096f2aa26c6d123611be.png)
, dokažite da vrijedi i jednakost
![aa^\prime = bb^\prime + cc^\prime](/media/m/5/e/c/5ecedc2912a631bd0dc1d7dc12e19cdf.png)
.
%V0
Ako za trokute s duljinama stranica $a$, $b$, $c$ i $a^\prime$, $b^\prime$, $c^\prime$ te nasuprotnim kutovima $\alpha$, $\beta$, $\gamma$ i $\alpha^\prime$, $\beta^\prime$, $\gamma^\prime$ vrijede jednakosti $\alpha + \alpha^\prime = \pi$ i $\beta = \beta^\prime$, dokažite da vrijedi i jednakost $aa^\prime = bb^\prime + cc^\prime$.
Izvor: Državno natjecanje iz matematike 2002