Question

如图,在斜边为AB的Rt△ABC中,由A作PA⊥平面ABC,AM⊥PB于M,AN⊥PC于N.?

(1)求证:BC⊥平面PAC;

(2)求证:PB⊥平面AMN.

Answer 1

证明:(1)∵PA⊥平面ABC,BC平面ABC,?

∴PA⊥BC.?

又∵AB为斜边,∴BC⊥AC.?

∵PA∩AC=A,∴BC⊥平面PAC.?

(2)∵BC⊥平面PAC,AN平面PAC,?

∴BC⊥AN.?

又∵AN⊥PC,且BC∩PC=C,?

∴AN⊥平面PBC.而PB平面PBC,?

∴AN⊥PB.?

又PB⊥AM,AM∩AN=A,∴PB⊥平面AMN.