选修4-1:几何证明选讲
如图,等边三角形
内接于圆
,以
为切点的圆
的两条切线交于点
,
交圆
于点
.
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/81a2034bb5884dbb8fb950289d74455b.png)
(1)求证:四边形
为菱形;
(2)若
,求等边三角形
的面积.
如图,等边三角形
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/b14f487803734a408ef11a0a24ab6cb3.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/d8c5684dcc7844df9dcdca6cc57f4e99.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/b9aaa9937d4947ac8e9fdebd4a81c03e.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/d8c5684dcc7844df9dcdca6cc57f4e99.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/0d182b60a23f4c2dbfd61088479c4ccb.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/7b36e47cb3b449099c42caa838f69826.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/d8c5684dcc7844df9dcdca6cc57f4e99.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/da51e06ba72147f0b7cf56d180428ad0.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/81a2034bb5884dbb8fb950289d74455b.png)
(1)求证:四边形
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/aae1360d683b409d8355fbf24ba4bec4.png)
(2)若
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/6be006f766f04a38b903a6283d9e7b33.png)
![](https://img.xkw.com/dksih/QBM/2016/10/18/1573077170307072/1573077331746816/STEM/b14f487803734a408ef11a0a24ab6cb3.png)
更新时间:2016-12-04 22:16:56
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【知识点】 几何证明选讲
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解答题-证明题
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【推荐1】如图,AB是圆O的直径,C,D是圆O上两点,AC与BD相交于点E,GC,GD是圆O的切线,点F在DG的延长线上,且
.求证:
(Ⅰ)D、E、C、F四点共圆; (Ⅱ)![](https://img.xkw.com/dksih/QBM/2013/8/30/1571334802751488/1571334808125440/STEM/27a75bc4eaa1470cbd90c3a339eaba96.png)
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(Ⅰ)D、E、C、F四点共圆; (Ⅱ)
![](https://img.xkw.com/dksih/QBM/2013/8/30/1571334802751488/1571334808125440/STEM/27a75bc4eaa1470cbd90c3a339eaba96.png)
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解答题-问答题
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适中
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名校
【推荐2】交中的新生小明同学非常喜欢数学,他在课外书上看到了一个有趣的定理——“中线长定理”:三角形两边的平方和等于第三边的一半与第三边上的中线的平方和的两倍.如图1,在
中,点D为BC中点,“中线长定理”即
.小明尝试对它进行证明,部分过程如下:
解:过点A作
于点E,如图2,在
中,
,
同理可得:
,
,
为证明的方便,不妨设
,
,
∴
……
(1)请你完成小明剩余的证明过程;
![](https://img.xkw.com/dksih/QBM/2021/9/22/2813365822922752/2814982063366144/STEM/4c257c9a-9135-4593-a527-bc99d8afc7bb.png?resizew=744)
理解运用:
(2)①在
中,点D为BC的中点,
,
,
,则
___________;
②如图3,
的半径为6,点A在圆内,且
,点B和点C在
上,且
,点E、F分别为AO,BC的中点,则EF的长为___________;
拓展延伸:
(3)小明解决上述问题后,联想到某课外书上的某题目:如图4,已知
的半径为
(圆心为原点O),以
为直角顶点的
的另两个顶点B,C都在
上,D为BC的中点,求AD长的最大值.请你利用上面的方法和结论,求出AD长的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51056472e8bc2ac5deb51b218b8e8dfd.png)
解:过点A作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e3f5dc11efe60b4fd9a13b1d6b83842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9681f9d187e18d23f604499a23727daf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e4ac0d942636f956146beb37e61ecbf.png)
同理可得:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/001e843fcc05f3e39aeea1e407f7630c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84bc23a1ff4ecafd57913425a1f1c14f.png)
为证明的方便,不妨设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3d6ee95e7130903b3a30b3bca0273c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5d70e5d13db498f1c8a2e017c56e58b.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce46caf1617393e9e4365d7b24a4df42.png)
(1)请你完成小明剩余的证明过程;
![](https://img.xkw.com/dksih/QBM/2021/9/22/2813365822922752/2814982063366144/STEM/4c257c9a-9135-4593-a527-bc99d8afc7bb.png?resizew=744)
理解运用:
(2)①在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a7b5adfcac0f46a4cd19da4ebb4a2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07140f277a35733d8c97577ccdd4e3ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9f63d194aa0d4091618b6f41f569ee2.png)
②如图3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e001897d70ceee81406f3b01d284b83b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c06154cae3bf7a8ce5a1e97a7380875.png)
拓展延伸:
(3)小明解决上述问题后,联想到某课外书上的某题目:如图4,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b947152b672cf56e5a8cde6800d71c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6646e758862a34d89ee14de8a1ea13b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
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