题目内容
(2004•丽水)已知⊙O1与⊙O2相切于点P,它们的半径分别为R、r.一直线绕P点旋转,与⊙O1、⊙O2分别交于点A、B(点P、B不重合),探索规律:(1)如图1,当⊙O1与⊙O2外切时,探求
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_ST/0.png)
(2)如图2,当⊙O1与⊙O2内切时,第(1)题探求的结论是否成立?为什么?
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_ST/images1.png)
【答案】分析:要求
与半径R、r之间的关系式,证明△O1AP∽△O2BP是关键,注意两圆的位置关系.
解答:
解:(1)当⊙O1与⊙O2外切时,
(3分)
证明:连接O1A,O2B
∵两圆外切,
∴O1、P、O2三点共线
∵△O1AP和△O2BP是等腰三角形,∠O1PA=∠BPO2,
∴∠O1AP=∠O2BP
∴△O1AP∽△O2BP
∴
;(4分)
(2)当⊙O1与⊙O2内切时,
仍然成立(2分)![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/images5.png)
证明:连接O1A,O2B,同理可证△PO1A∽△PO2B,
∴
仍然成立.(3分)
(注:能指出当动直线AB经过两圆的圆心时,PA=2R,PB=2r,∴
,奖励1分.)
点评:本题考查了两圆的位置关系,相似三角形的判定和性质,是一个探究性的题目.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/0.png)
解答:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/images1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/1.png)
证明:连接O1A,O2B
∵两圆外切,
∴O1、P、O2三点共线
∵△O1AP和△O2BP是等腰三角形,∠O1PA=∠BPO2,
∴∠O1AP=∠O2BP
∴△O1AP∽△O2BP
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/2.png)
(2)当⊙O1与⊙O2内切时,
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/images5.png)
证明:连接O1A,O2B,同理可证△PO1A∽△PO2B,
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/4.png)
(注:能指出当动直线AB经过两圆的圆心时,PA=2R,PB=2r,∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232258943918270/SYS201310212322589439182017_DA/5.png)
点评:本题考查了两圆的位置关系,相似三角形的判定和性质,是一个探究性的题目.
![](http://thumb2018.1010pic.com/images/loading.gif)
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