题目内容
(2011•安庆一模)我们把1°的圆心角所对的弧叫做l°的弧.则圆心角AOB的度数等于它所对的弧AB的度数记为:∠AOB=![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_ST/0.png)
问题(1):如图2,⊙0的两条弦AB、CD相交于圆内一点P,求证:∠APC=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_ST/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_ST/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_ST/3.png)
问题(2):如图3,⊙0的两条弦AB、CD相交于圆外一点P.问题(1)中的结论是否成立?如果成立,给予证明;如果不成立,写出一个类似的结论(不要求证明)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_ST/images4.png)
【答案】分析:(1)根据圆周角的度数等于其所对的弧的度数的一半,得∠AOB![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/0.png)
,连BC,可证得∠APC=
(
+
);
(2)问题(1)中的结论不成立.类似的结论为:∠BPC=
(
-
).
解答:证明:∵∠APB=
∠AOB,又∠AOB![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/9.png)
,
∴即圆周角的度数等于其所对的弧的度数的一半.(4分)
(1)连BC,则∠APC=∠PCB+∠PBC
∠PCB的度数等于弧BD的度数的一半,∠PBC的度数等于弧AC的度数的一半,
∠APC=
(
+
);
(2)问题(1)中的结论不成立.(11分)
类似的结论为:∠BPC=
(
-
).
点评:本题考查了圆周角定理以及圆心角、弧、弦之间的关系,是基础知识要熟练掌握.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/4.png)
(2)问题(1)中的结论不成立.类似的结论为:∠BPC=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/5.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/6.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/7.png)
解答:证明:∵∠APB=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/8.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/9.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/10.png)
∴即圆周角的度数等于其所对的弧的度数的一半.(4分)
(1)连BC,则∠APC=∠PCB+∠PBC
∠PCB的度数等于弧BD的度数的一半,∠PBC的度数等于弧AC的度数的一半,
∠APC=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/11.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/12.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/13.png)
(2)问题(1)中的结论不成立.(11分)
类似的结论为:∠BPC=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/14.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/15.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021233102949381488/SYS201310212331029493814022_DA/16.png)
点评:本题考查了圆周角定理以及圆心角、弧、弦之间的关系,是基础知识要熟练掌握.
![](http://thumb2018.1010pic.com/images/loading.gif)
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