Državno natjecanje 2004 SŠ4 2
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arhiva1. travnja 2012. unutar troukta
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
s duljinama stranica
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
i odgovarajucim kutevima
![\alpha, \beta, \gamma](/media/m/2/8/6/286052c3ef6a627f9d4f5349ddaf2ba7.png)
postoje tocke
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
i
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
takve da vrijedi
![\angle BPC = \angle CPA = \angle APB = 120^\circ](/media/m/4/7/f/47fb7421391d900d5c81719b20b8282c.png)
,
![\angle BQC = 60^\circ + \alpha, \angle CQA = 60^\circ + \beta, \angle AQB = 60^\circ + \gamma](/media/m/b/7/4/b745e50e604066c5516077e9452783f5.png)
.
dokazite da vrijedi jednakost
%V0
unutar troukta $ABC$ s duljinama stranica $a, b, c$ i odgovarajucim kutevima $\alpha, \beta, \gamma$ postoje tocke $P$ i $Q$ takve da vrijedi
$\angle BPC = \angle CPA = \angle APB = 120^\circ$,
$\angle BQC = 60^\circ + \alpha, \angle CQA = 60^\circ + \beta, \angle AQB = 60^\circ + \gamma$.
dokazite da vrijedi jednakost
$(|AP| + |BP| + |CP|)^3\cdot|AQ|\cdot|BQ|\cdot|CQ| = (abc)^2$
Izvor: Državno natjecanje iz matematike 2004