(1)证明:如图一,∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
,F分别是AB,AC,BC边的中点,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230036480201350.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
F∥AC且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
F =A
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
F∥AB且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
F =A
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
,
∴∠B
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
F=∠BAC,∠C
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
F=∠BAC,
∴∠B
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
F=∠C
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
F
∵点D和点E分别为两个半圆圆弧的中点,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
F =A
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
E,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
F =A
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
D, ……………………….2分
∠B
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
D =90°,∠C
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
E =90°,
∴∠B
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
D=∠C
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
E.
∴∠D
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
F=∠F
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
E.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648035574.gif)
. ………………………….3分
(2)解:如图二,延长CA至G,使AG=AQ,连接BG、AE.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230036486901938.gif)
∵点E是半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
圆弧的中点,
∴AE=CE=3
∵AC为直径
∴∠AEC=90°,
∴∠ACE=∠EAC =45°,AC=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648737475.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648753241.gif)
,
∵AQ是半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
的切线,
∴CA⊥AQ,∴∠CAQ=90°,
∴∠ACE=∠AQE=45°,∠GAQ="90°"
∴AQ=AC=AG=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648753241.gif)
同理:∠BAP=90°,AB=AP=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648784234.gif)
∴CG=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648815251.gif)
,∠GAB=∠QAP
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648815543.gif)
. ……………………..5分
∴PQ=BG
∵∠ACB=90°,
∴BC=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648831468.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648846241.gif)
∴BG=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648862381.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648144272.gif)
∴PQ=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648144272.gif)
. …………………..6分
(3) 证法一:如图三,设直线FA与PQ的垂足为M,过C作CS⊥MF于S,过B作BR⊥MF于R,连接DR、AD、DM.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230036489242251.gif)
∵F是BC边的中点,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648940434.gif)
.
∴BR=CS,
由(2)已证∠CAQ="90°," AC=AQ,
∴∠2+∠3=90°
∵FM⊥PQ, ∴∠2+∠1=90°,
∴∠1=∠3,
同理:∠2=∠4,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648956565.gif)
,
∴AM=CS,
∴AM=BR,
同(2)可证AD=BD,∠ADB=∠ADP=90°,
∴∠ADB=∠ARB="90°," ∠ADP=∠AMP=90°
∴A、D、B、R四点在以AB为直径的圆上,A、D、P、M四点在以AP为直径的圆上,
且∠DBR+∠DAR=180°,
∴∠5=∠8, ∠6=∠7,
∵∠DAM+∠DAR=180°,
∴∠DBR=∠DAM
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003648971547.gif)
,
∴∠5=∠9,
∴∠RDM=90°,
∴∠5+∠7=90°,
∴∠6+∠8=90°,
∴∠PAB=90°,
∴PA⊥AB,又AB是半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
直径,
∴PA是半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
的切线. ……………………..8分
证法二:假设PA不是是半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
的切线,如图四,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230036490342801.gif)
过点A作半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
的切线交BD的延长线于点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649065202.gif)
,则点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649065202.gif)
异于点P,连结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649080357.gif)
,设直线FA与PQ的垂足为M,直线FA与
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649080357.gif)
的交点为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649112327.gif)
.延长AF至N,使得AF=FN,连结BN,CN,由于点F是BC中点,所以四边形ABNC是平行四边形.
易知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649143598.gif)
,
∵AQ是半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647973216.gif)
的切线,
∴∠QAC=90°,同理
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649174462.gif)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649190624.gif)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649205530.gif)
.
由(2)可知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649221544.gif)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649236558.gif)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649252528.gif)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649268475.gif)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649268624.gif)
.
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649283649.gif)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649314509.gif)
.
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649330428.gif)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649346421.gif)
,
∴ 过点Q有两条不同的直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649080357.gif)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003649377243.gif)
同时与AF垂直.
这与在平面内过一点有且仅有一条直线与已知直线垂直相矛盾,
因此假设错误.所以PA是是半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823003647957214.gif)
的切线.