题目内容
(满分14分)如图,⊙O的半径为1,点P是⊙O上一点,弦AB垂直平分线段OP,点D是劣弧AB上任一点(与端点A、B不重合),DE⊥AB于点E,以点D为圆心、DE长为半径作⊙D,分别过点A、B作⊙D的切线,两条切线相交于点C.
(1)求弦AB的长;
(2)判断∠ACB是否为定值,若是,求出∠ACB的大小;否则,请说明理由;
(3)记△ABC的面积为S,若
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(1)
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(2)略
(3)
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(1)连接OA,取OP与AB的交点为F,则有OA=1.
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∵弦AB垂直平分线段OP,∴OF=
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在Rt△OAF中,∵AF=
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∴AB=2AF=
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(2)∠ACB是定值.
理由:由(1)易知,∠AOB=120°,
因为点D为△ABC的内心,所以,连结AD、BD,则∠CAB=2∠DAE,∠CBA=2∠DBA,
因为∠DAE+∠DBA=
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(3)记△ABC的周长为l,取AC,BC与⊙D的切点分别为G,H,连接DG,DC,DH,
则有DG=DH=DE,DG⊥AC,DH⊥BC.
∴
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=
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∵
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∵CG,CH是⊙D的切线,∴∠GCD=
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∴在Rt△CGD中,CG=
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又由切线长定理可知AG=AE,BH=BE,
∴l=AB+BC+AC=2
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