如圖,設(shè)AB,CD為⊙O的兩直徑,過(guò)B作PB垂直于AB,并與CD延長(zhǎng)線相交于點(diǎn)P,過(guò)P作直線與⊙O分別交于E,F(xiàn)兩點(diǎn),連結(jié)AE,AF分別與CD交于G、H
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240212478025889.png)
(Ⅰ)設(shè)EF中點(diǎn)為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247818336.png)
,求證:O、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247818336.png)
、B、P四點(diǎn)共圓
(Ⅱ)求證:OG =OH.
(Ⅰ)詳見(jiàn)解析;(Ⅱ)詳見(jiàn)解析.
試題分析:利用對(duì)角互補(bǔ)得到四點(diǎn)共圓,利用相似得到邊長(zhǎng)相等.
試題解析:證明:(Ⅰ)
易知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247849884.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247865603.png)
四點(diǎn)共圓. 3分
(Ⅱ)由(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247880708.png)
過(guò)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247896302.png)
作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247911585.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247927351.png)
,交
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247943396.png)
于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247958343.png)
連結(jié)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247974662.png)
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248005438.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248021407.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248036774.png)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248036791.png)
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248052633.png)
四點(diǎn)共圓. 6分
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248067928.png)
,由此
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248083450.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248099410.png)
, 8分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248099339.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248114383.png)
的中點(diǎn),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021247958343.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824021248145420.png)
的中點(diǎn),所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240212481611106.png)
,所以O(shè)G ="OH" 10分
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