过三角形ABC所在平面a外一点P,作PO垂直于a,垂足为O,连接PA,PB,PC
若PA垂直PB PB垂直PC PC垂直PA ,则点o是三角形ABC的______心?
人气:459 ℃ 时间:2019-12-13 17:45:52
解答
O是△ABC的垂心.
∵PA ⊥ PB,PA ⊥ PC,
∴PA ⊥ 平面PBC,
∴PA ⊥ BC.
又∵PO ⊥ 平面ABC,
∴OA是PA在平面ABC内的射影.
∴OA ⊥ BC (三垂线定理).
同理,OB ⊥ AC,OC ⊥ AB.
即O是△ABC的垂心.
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