在
中,
,
,
、
分别平分
和
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acdbac251ca6b76a166c1242e71df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d320f180419175d75eebc618cc458b39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a55a1a244f81097e05e715b69580faa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/936f5f26aac7197d42a41081fb1fda0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad6891e11d452f7da4939010ebc0aa88.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/3/d801d031-22dc-48d4-8ca7-7ea1cef6fffc.png?resizew=84)
2022九年级上·全国·专题练习 查看更多[1]
(已下线)专题24 定点定长构造辅助圆-【微专题】2022-2023学年九年级数学上册常考点微专题提分精练(人教版)
更新时间:2022-11-29 15:25:09
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解答题-证明题
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名校
【推荐1】如图,在
中,
分别是
的中点,
.
是菱形;
(2)若
,求四边形
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ce7f6d278f2ef7a193b7eed7be6b3c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbac29f283fcfdb5428740948e0c3a99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86cae7cc6665dc8edfb9060075b3afb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b74e943d234d0985873a10fa2eba54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041e4132c761a05c17d9d54ae3e41151.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f40ed480902c26eade58471d094eaa92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041e4132c761a05c17d9d54ae3e41151.png)
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【推荐2】如图,矩形
中
,连接
.
(1)求证:四边形
是平行四边形;
(2)取
的中点
并连接,若
,试求线段
的长度.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ce1475f537b4ad21775bfaa16daa0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e825678030322bf7eb2f0249bf28ae25.png)
(1)求证:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2330c01a4d2b5b20f106e3e48834d5c0.png)
(2)取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19097651607095bc2bf9298bb964c392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8368e547d5e1f262de0119ca7c98634a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://img.xkw.com/dksih/QBM/2020/8/25/2535412237860864/2541153522688000/STEM/c65aac9d62164dbda9991eba2620f3d9.png?resizew=192)
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【推荐1】我们知道,直线与圆有三种位置关系:相交、相切、相离.当直线与圆有两个公共点(即直线与圆相交)时,这条直线就叫做圆的割线.割线也有一些相关的定理.比如,割线定理:从圆外一点引圆的两条割线,这一点到每条割线与圆交点的距离的积相等.下面给出了不完整的定理“证明一”,请补充完整.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/9/002aeb80-71a9-41c0-8d46-9e325361fe4e.png?resizew=385)
已知:如图①,过
外一点
作
的两条割线,一条交
于
、
点,另一条交
于
、
点.
求证:
.
证明一:连接
、
,
∵
和
为
所对的圆周角,∴______.
又∵
,∴______,∴______.
即
.
研究后发现,如图②,如果连接
、
,即可得到学习过的圆内接四边形
.那么或许割线定理也可以用圆内接四边形的性质来证明.请根据提示,独立完成证明二.
证明二:连接
、
,
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/9/002aeb80-71a9-41c0-8d46-9e325361fe4e.png?resizew=385)
已知:如图①,过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a5f6255503de9bcc36f483093b366e2.png)
证明一:连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/194741f4d2ae7ee44cafca780361446a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bd6ffb78dad3375efa3b08ab518553d.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/651c6ad718e8cf15ad68d2cdbc5dad57.png)
即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a5f6255503de9bcc36f483093b366e2.png)
研究后发现,如图②,如果连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/520f8abda6a85e7ef6f281fc2df853fa.png)
证明二:连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
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【推荐2】数学思想方法对于数学学习至关重要,不仅可以建立正确的数学思维方式,提高解决问题的能力,还可以培养创造性思维和逻辑思维,增强逻辑思维能力和创新能力.在研究一条弧所对的圆周角和圆心角之间的关系时就涉及了完全归纳法.完全归纳法是把要研究的某类事物的所有情况,先逐一加以讨论,再概括得出一般结论.有时也可以把所有情况分成几类,对每类加以讨论,再概括得出一般情况.在利用完全归纳法证明的过程中,应明确为什么要分情况证明,而且要分得准确,要不重不漏.为了证明同弧所对的圆周角的度数等于这条弧所对的圆心角的度数的一半,张老师引导学生画图,在
中任取一个圆周角
,由于点A的位置不同,会出现三种情况,
(1)如图1,圆心O在
的一边上;
(2)如图2,圆心O在
的内部;
(3)如图3,圆心O在
的外部.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/17/44eb96d4-fa91-46ec-8276-4fc398e565a0.png?resizew=408)
张老师引导学生分析第(1)种情况,如图1.∵
,∴
.
∵
是
的外角,∴
.∴
.
对于第(2)(3)种情况,张老师引导学生们分析,通过添加辅助线,可以将它们转化为第(1)种情况,从而得到相同的结论.请你完成第(2)(3)种情况的证明.
(1)如图2,连接AO并延长AO交
于点D,求证:
;
(2)如图3,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
(1)如图1,圆心O在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
(2)如图2,圆心O在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
(3)如图3,圆心O在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/17/44eb96d4-fa91-46ec-8276-4fc398e565a0.png?resizew=408)
张老师引导学生分析第(1)种情况,如图1.∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a23f35ebcd9799d82c1e41c09781a4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1393308bc68322f50fa29702dd412263.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d22df2977de56cc69be0c1e847653d7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bbf9680f74a9ac5d934304654ce2771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7db4c210dc258530e60cfeeefda24f0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b18abbe2ccf6c6c6a4efb59fb887dd6e.png)
对于第(2)(3)种情况,张老师引导学生们分析,通过添加辅助线,可以将它们转化为第(1)种情况,从而得到相同的结论.请你完成第(2)(3)种情况的证明.
(1)如图2,连接AO并延长AO交
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b18abbe2ccf6c6c6a4efb59fb887dd6e.png)
(2)如图3,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b18abbe2ccf6c6c6a4efb59fb887dd6e.png)
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