Državno natjecanje 1997 SŠ3 3
Dodao/la:
arhiva1. travnja 2012. Neka su u tetraedru
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površina strana
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,
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,
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i
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redom jednake
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,
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,
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,
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, a prostorni kut između strana
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i
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jednak
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, odnosno
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između
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i
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. Dokažite da je
%V0
Neka su u tetraedru $ABCD$ površina strana $ABD$, $ACD$, $BCD$ i $BCA$ redom jednake $S_1$, $S_2$, $Q_1$, $Q_2$, a prostorni kut između strana $ABD$ i $ACD$ jednak $\alpha$, odnosno $\beta$ između $BCD$ i $BCA$. Dokažite da je $$S_1^2 + S_2^2 - 2S_1S_2\cos \alpha = Q_1^2 + Q_2^2 - 2Q_1Q_2\cos \beta \text{.}$$
Izvor: Državno natjecanje iz matematike 1997