(1)基础应用:如图1,在△ABC中,AB=5,AC=7,AD是BC边上的中线,延长AD到点E使DE=AD,连接CE,把AB,AC,2AD利用旋转全等的方式集中在△ACE中,利用三角形三边关系可得AD的取值范围是 ;
(2)推广应用:应用旋转全等的方式解决问题如图2,在△ABC中,AD是BC边上的中线,点E,F分别在AB,AC上,且DE⊥DF,求证:BE+CF>EF;
(3)综合应用:如图3,在四边形ABCD中,AB=AD,∠B+∠ADC=180°且∠EAF=
∠BAD,试问线段EF、BE、FD具有怎样的数量关系,并证明.
(2)推广应用:应用旋转全等的方式解决问题如图2,在△ABC中,AD是BC边上的中线,点E,F分别在AB,AC上,且DE⊥DF,求证:BE+CF>EF;
(3)综合应用:如图3,在四边形ABCD中,AB=AD,∠B+∠ADC=180°且∠EAF=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/30/7a52d5e4-fdba-4903-b725-83953a1ba779.png?resizew=423)
20-21七年级下·四川达州·期末 查看更多[7]
四川省达州市开江县2020-2021学年七年级下学期期末数学试题(已下线)专题12.34 作辅助线证明三角形全等-倍长中线(培优篇)(专项练习)-2022-2023学年八年级数学上册基础知识专项讲练(人教版)江西省吉安市永丰县2021-2022学年七年级下学期期末数学试题(已下线)专题强化训练一 全等三角形中的辅助线(模型)问题-2022-2023学年七年级数学下册《考点·题型·技巧》精讲与精练高分突破系列(北师大版)(已下线)重难点03 全等三角形(4种模型2种添加辅助线方法)-【满分全攻略】2022-2023学年七年级数学下学期核心考点+重难点讲练与测试(沪教版)(已下线)专题12.21 全等三角形几何模型(倍长中线)(分层练习)-2023-2024学年八年级数学上册基础知识专项突破讲与练(人教版)(已下线)专题1.21 全等三角形几何模型-倍长中线(分层练习)-2023-2024学年八年级数学上册基础知识专项突破讲与练(苏科版)
更新时间:2021-08-08 15:16:49
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相似题推荐
解答题-证明题
|
较难
(0.4)
【推荐1】定义:有一组对边相等且这一组对边所在直线互相垂直的凸四边形叫做“等垂四边形”.如图
,四边形
中,
,
,四边形
即为等垂四边形,其中相等的边
,
称为腰,另两边
,
称为底.
(1)如图
,
与
都是等腰直角三角形,
,
.求证:四边形
是“等垂四边形”.
【拓展探究】
(2)如图
,四边形
是“等垂四边形”,
,点
,
分别是
,
的中点,连接
.已知腰
,求
的长.
【综合运用】
(3)如图
,四边形
是“等垂四边形”,腰
,底
,则较短的底
长的取值范围为_________.(直接写出答案)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c220eadc312101e2fb89dfe920f7b30d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a77e3c1c236141d6118429fade0a9b9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(1)如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2689f0ce5ab3467d8214794d8acb2bd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc75953bf5dcfa4af308c34bf9952d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d30bee15912841dbe5780259c3ddc52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8a4fc8a8b67d26aaf8a723313b14a7b.png)
【拓展探究】
(2)如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aade3d5cbfd7ab6a8595b29716a52a0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](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/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
【综合运用】
(3)如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51e0f9d0d28bfb81ad132e0064402573.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7788830ed1cb3b9c5988f70f43595f2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
您最近一年使用:0次
解答题-证明题
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【推荐2】【发现问题】数学兴趣小组在活动时,老师提出了这样的一个问题:
如图①,在
中,
,
,第三边上的中线
,则
的取值范围是____.
(1)如图②,延长
至点
,使得
,连结
,根据“
”可以判定
__________,得出
.在
中,
,
,
,故中线
的长x的取值范围是_______.
【活动经验】当条件中出现“中点”,“中线”等条件时,可以考虑将中线延长一倍,构造全等三角形,把分散的已知条件和所求的问题集中到同一个三角形中,进而解决问题,这种作辅助线的方法叫做“倍长中线”法.
【问题解决】(2)如图③,已知
,
,
,连接
和
,点
是
的中点,连接
.求证:
.小明发现,如图④,延长
至点
,使
,连接
,通过证明
,可推得
.
下面是小明的部分证明过程:
证明:延长
至点
,使
,连接
,
∵点
是
的中点,
∴
.
∵
,
,
∴
,
∴
,
,
∴
,
.
请你补全余下的证明过程.
【问题拓展】(3)如图⑤,在
和
中,
,
,
,点M,N分别是
和
的中点.若
,
,则MN的取值范围是 .
如图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/4494a85de0be0b97a69348115aef8513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2633a6f46b0847158eebb05c743561aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)如图②,延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/429aee631c9e759b7788a83610d4df7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eb97aff0960e2640314888a38e7169c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5a290f047f50481318d040c604d72f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a75544962e7df0b344e5a5a992cd203.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5733d12d191a3ec7bb2416af1ff5a5ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2431595096e6c55036353c265654e5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd1b65108461a6941a5a7eee52fa7548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4494a85de0be0b97a69348115aef8513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46df601a5f58bcf4ec6c5ccb2fb4bb9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
【活动经验】当条件中出现“中点”,“中线”等条件时,可以考虑将中线延长一倍,构造全等三角形,把分散的已知条件和所求的问题集中到同一个三角形中,进而解决问题,这种作辅助线的方法叫做“倍长中线”法.
【问题解决】(2)如图③,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d212c1709b8e72a055cf1b5381ef64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb9e83916bac21e85ca878710844cbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/905f0c64cc1ce21fa230b6e53bde940f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f2d42ce53fbf68efbd860e91d440b44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a21897349d3d7c94419692106887153.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82d07a1e3479344c2a1f73b002b5c8de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93b4cd4409fcdf0f3bacc7f6629d143e.png)
下面是小明的部分证明过程:
证明:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f2d42ce53fbf68efbd860e91d440b44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a21897349d3d7c94419692106887153.png)
∵点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/108ebf3215a544f65ff55283bda648b9.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb430059ddfa0934e039dbd26836886b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6842ddc34d3c6f2413551b2ac05ae94f.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91b803a5dddecbc38cd6719152642f94.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da5a8cd3789ec61ac02b11e4481d4e2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60e6cfd0ea335c14fef466ae96d6bee8.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea5bf3ba1d4025afc225eec5c298653.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07f69d4fec03d944448740b93cf3172c.png)
请你补全余下的证明过程.
【问题拓展】(3)如图⑤,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48a277db452e76240ec83ec6a2864bdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49cb6392b1f859954303066037c5f5fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02cd561f48b21006701b8dabaa6f95d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a028fd5b281ef168702a803baca6e8.png)
您最近一年使用:0次
解答题-证明题
|
较难
(0.4)
【推荐1】综合与实践
问题情境:
在
和
中,
,
.将
的顶点
放在
底边
的中点处,
的顶点
与
底边
的中点重合.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/18/df709f4e-e004-4f30-8f75-361072f0f728.png?resizew=500)
猜想证明:
(1)如图1,
与
的交点记为
,
与
的交点记为
,试判断四边形
的形状,并说明理由;
问题解决:
将
绕点
旋转,
边
与
交于点
.
(2)如图2,在
旋转过程中,当
平分
时,求线段
的长;
(3)如图3,在
旋转过程中,当
时,直接写出线段
的长.
问题情境:
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e64e83a70cd8b540f68b8548f5ba7ee1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33a8077d9f92002c56072326bcf93025.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/18/df709f4e-e004-4f30-8f75-361072f0f728.png?resizew=500)
猜想证明:
(1)如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04a6fa5218491da01843cdfde0c3dcbd.png)
问题解决:
将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
(2)如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c962fe4f47732b8e6e83d17ff2b9af13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
(3)如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/100b3e7dcd5f34c5234971cd75bff138.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7e4fa04825ac7d071968056322d88be.png)
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真题
【推荐2】在数学兴趣小组活动中,小亮进行数学探究活动.
(1)
是边长为3的等边三角形,E是边
上的一点,且
,小亮以
为边作等边三角形
,如图1,求
的长;
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/21/0cfdb694-b8fb-491e-b50a-e883ee2ff999.png?resizew=286)
(2)
是边长为3的等边三角形,E是边
上的一个动点,小亮以
为边作等边三角形
,如图2,在点E从点C到点A的运动过程中,求点F所经过的路径长;
(3)
是边长为3的等边三角形,M是高
上的一个动点,小亮以
为边作等边三角形
,如图3,在点M从点C到点D的运动过程中,求点N所经过的路径长;
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/21/ccba8556-84ff-4277-a0db-83b9ba1ccb0c.png?resizew=337)
(4)正方形
的边长为3,E是边
上的一个动点,在点E从点C到点B的运动过程中,小亮以B为顶点作正方形
,其中点F、G都在直线
上,如图4,当点E到达点B时,点F、G、H与点B重合.则点H所经过的路径长为______,点G所经过的路径长为______.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1364213f546b37f8764ddcb59e36ae4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c0bfeadcf17b2a45896071f07a4a5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/21/0cfdb694-b8fb-491e-b50a-e883ee2ff999.png?resizew=286)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c0bfeadcf17b2a45896071f07a4a5a.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212a67f115d1cbe69f100b489babe5f8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/21/ccba8556-84ff-4277-a0db-83b9ba1ccb0c.png?resizew=337)
(4)正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db7e7056e5433b965138327ab62dc65a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
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【推荐1】课外兴趣小组活动时,老师提出了如下问题:如图1,
中,若
,求
边上的中线
的取值范围.小明在组内经过合作交流,得到了如下的解决方法:如图1所示,延长
到点
,使
,连接
.请根据小明的思路继续思考:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/22/7bdbf9e5-40ae-4f78-a41d-da0fc360647d.png?resizew=414)
(1)由已知和作图能证得
,得到
,在
中求得
的取值范围,从而求得
的取值范围是______________.
方法总结:上述方法我们称为“倍长中线法”.“倍长中线法”多用于构造全等三角形和证明边之间的关系;
(2)如图2,
是
的中线,
,试判断线段
与
的数量关系,并加以证明;
(3)如图3,在
中,
是
的三等分点.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fed6f0c9013332e8d41bf651ee75b915.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ad98ad714864041a632ca949308e417.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/22/7bdbf9e5-40ae-4f78-a41d-da0fc360647d.png?resizew=414)
(1)由已知和作图能证得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e01ccfa50b9cfcffc453c323d028571e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7595bdaea3dfa548c0fcfe3708387476.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e97392ff60fac8a261c6eab71bba028b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
方法总结:上述方法我们称为“倍长中线法”.“倍长中线法”多用于构造全等三角形和证明边之间的关系;
(2)如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ca1627a0f21b9983e97651143e53214.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
(3)如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff03a300a077600a87f93aed93a29ce.png)
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名校
【推荐2】阅读下面的题目及分析过程.
已知:如图1,点E是
的中点,点A在
上,且
.说明:
.
分析:说明两个角相等,常用的方法是应用全等三角形或等腰三角形的性质. 观察本题中说明的两个角,它们既不在同一个三角形中,而且它们所在两个三角形也不全等.因此,要说明
,必须添加适当的辅助线,构造全等三角形或等腰三角形,现在提供两种添加辅助线的方法如下:如图2,过点C作
,交
的延长线于点F;如图3,延长
至点M,使
,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/2/93489c2f-e6c2-4573-8fd6-1ec5c52dc9aa.png?resizew=697)
(1)请从以上两种添加辅助线方法中选择一种完成上面的说理过程.
(2)反思应用:如图4,点B是
的中点,
于点B.请类比(1)中解决问题的思想方法,添加适当的辅助线,判断线段
与
之间的大小关系,并说明理由.
已知:如图1,点E是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfb27272b4eb9ed057dd3b4d40697e7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e0372ab874b7a22402858bdbbf0f63b.png)
分析:说明两个角相等,常用的方法是应用全等三角形或等腰三角形的性质. 观察本题中说明的两个角,它们既不在同一个三角形中,而且它们所在两个三角形也不全等.因此,要说明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e0372ab874b7a22402858bdbbf0f63b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b00a5239269d72d6f9632e7d77347106.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102fed4da9bd481ea6e0cd8ebe9c8bbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/2/93489c2f-e6c2-4573-8fd6-1ec5c52dc9aa.png?resizew=697)
(1)请从以上两种添加辅助线方法中选择一种完成上面的说理过程.
(2)反思应用:如图4,点B是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13d28cb7181257cf732af4b615fc47d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2957078bc951e47bd985626d19d42404.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
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【推荐3】【阅读理解】课外兴趣小组活动时,老师提出了如下问题:
如图,
中,
,
,求
边上的中线
的取值范围,经过组内合作交流.小明得到了如下的解决方法:延长
到点
,使![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ad98ad714864041a632ca949308e417.png)
请根据小明的方法思考:
(1)求得
的取值范围是___________;
【问题解决】请利用上述方法(倍长中线)解决下列三个问题
如图,已知
,
,
,
为
的中点.
(2)如图1,若
,
,
共线,求证:
平分
;
(3)如图2,若
,
,
不共线,求证:
;
(4)如图3,若点
在
上,记锐角
,且
,则
的度数是___________(用含
的代数式表示).
如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54275b7e571660d0a9e0370fbfe5050b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1682d306c38087d9e6f7efb9cec596a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ad98ad714864041a632ca949308e417.png)
请根据小明的方法思考:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/25/216900bb-adbf-485a-b83c-912df4be6fdd.png?resizew=148)
(1)求得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
【问题解决】请利用上述方法(倍长中线)解决下列三个问题
如图,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aabd1d9b7e7f7ecb788ff542218af2b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d831c9ba6d18774540e2aaa29132fec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/25/576eadbb-598b-470e-8f22-c7a6fb37f15b.png?resizew=695)
(2)如图1,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
(3)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71be6f57fdb788b6fe4a89efababd279.png)
(4)如图3,若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f597657b1072e1d8aa1c754975c2c4c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f77f3de68b497dfdb1e122ec9cc8832.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b8a8bede44a7720564504891f7982e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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