已知三棱锥 S-ABC 的底面是正三角形，A 点在侧面 SBC 上的射影 H 是△SBC 的垂心，二面角&nb... 已知三棱锥S—ABC的底面是正三角形，A点在侧面SBC上的射...
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由三垂线定理可知SC⊥AE，SC⊥AB，所以SC⊥面ABE．
设S在面ABC内射影为O，则SO⊥面ABC．
由三垂线定理之逆定理，可知CO⊥AB于F．
同理，BO⊥AC，故O为△ABC的垂心．
又因为△ABC是等边三角形，故O为△ABC的中心，从而SA=SB=SC=2．
因为CF⊥AB，CF是EF在面ABC上的射影，由三垂线定理，EF⊥AB．
所以，∠EFC是二面角H-AB-C的平面角，故∠EFC=30°，
∴OC=SCcos60°=1，
∴SO=
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