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一、一个有用的结论三棱锥A-EFG的三条侧棱AE,AF,AG两两互相垂直,三侧面与底面所成二面角分别记为α,β,γ,则有cos~2α+cos~2β+cos~2γ=1．证明：如图1,O是A在底面EFG上的射影,连接EO,FO,GO并延长,分别交三边于P,Q,R,连接AP,AQ,AR．∵AE,AF,AC两两垂直,∴AE⊥面AFG,因此AE⊥FG．
A. A useful conclusion The three side edges AE, AF, and AG of the triangular pyramid A-EFG are perpendicular to each other. The dihedral angles formed by the three sides and the bottom surface are denoted as α, β, γ, respectively, and there are cos~2α+cos. ~2β+cos~2γ=1. Proof: As shown in Figure 1, O is the projection of A on the bottom EFG, connecting EO, FO, GO and extending, respectively, to three sides on P, Q, R, connecting AP, AQ, AR. ∵AE, AF, AC two vertical, ∴AE ⊥ face AFG, so AE ⊥ FG.