1 . 如图,平行四边形
的对角线
、
交于点
,
平分
交
于点
,交
于点
,
,
,连接
.下列结论:①
;②
平分
;③
;④
.其中正确的个数有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](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/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72c8ba096171441d3178bd110785cf37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ec435aa1401dbce7863b531bf2f3e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f74a61581d965458316280968241e4c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6489e4f301f536d848278e96af474363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d97dc3b752832906de41447bb58a341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7d19f963f7c803bcfcbb265ec6d4304.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1625cb560f6b29ee8e86f491970472c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9903a185d30b887a5bc204df3ad59b1.png)
A.1个 | B.2个 | C.3个 | D.4个 |
您最近一年使用:0次
2 . 如图,在
中,
,点D,E,F分别为
,
,
的中点.
是菱形;
(2)若
,
,求四边形
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48a61d8bedef61745ee4fd89a8c0c119.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ebb33adb2310a6e03918761e68204a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48a61d8bedef61745ee4fd89a8c0c119.png)
您最近一年使用:0次
3 . 定义.对于一个四边形,我们把依次连结它的各边中点得到的新四边形叫做原四边形的“中点四边形”,如果原四边形的中点四边形是个正方形,我们把这个原四边形叫做“中方四边形”.
概念理解:下列四边形中一定是“中方四边形”的是______________
A.平行四边形 B.矩形 C.菱形; D.正方形.
性质探究:如图1,四边形
是“中方四边形”,观察图形,写出关于四边形
性质的一条结论:___________
问题解决:如图2,以锐角
的两边
为边长,分别向外侧作正方形
和正方形
,连结
.
是“中方四边形”;
拓展应用:如图3,已知四边形
是“中方四边形”,M,N分别是
的中点.
(2)若
,则
_________;
(3)若
的最小值是2,则
的长度为_________;
概念理解:下列四边形中一定是“中方四边形”的是______________
A.平行四边形 B.矩形 C.菱形; D.正方形.
性质探究:如图1,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
问题解决:如图2,以锐角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17487118ab7a90fc8c91bf0870f3289e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad3a079cfdcca9acdacecbf08f9f78cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff7a0bfc593a8a33b6cade6ba213904c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a58bcffc28a1d6bfdfd04a4abed39d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/693cd6179b2a92f03153ce12a0e86b95.png)
拓展应用:如图3,已知四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdcf86f7cf381c563ad80af86feeed83.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/056c2272e0d10d6dd9706e6324d8e62d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c978d92edf0c4c1ef8620c17df75d35e.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1aac055263010139f6eb4df68f9d619.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
您最近一年使用:0次
4 . 如图,在矩形
中,
,
,P是边
上的任意一点,连接
,
,E,F,G分别是
,
,
的中点.
与
的数量关系为________;位置关系为________;
(2)试猜想:当点P位于什么位置时,四边形
是菱形?并证明猜想的正确性;
(3)若(2)中菱形
为正方形,直接写出a与b之间的数量关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d42e97eee705d164e6ac6de9ecd6d1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c52b857c7ca55dfa6da108df1d3cee2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4505508b3e36db64a207dcdaf8eb22dc.png)
(2)试猜想:当点P位于什么位置时,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0363419eabe432409a7f04598450a396.png)
(3)若(2)中菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0363419eabe432409a7f04598450a396.png)
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2024八年级下·全国·专题练习
5 . 如图,在
中,点
,
分别为
,
的中点,延长
至点
,使得
,连接
,
.
;
(2)求证:四边形
是平行四边形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b00a5239269d72d6f9632e7d77347106.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d3971a9519246550390de4bd599617d.png)
(2)求证:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10af0a2e32fb3f3af9664617ac05b669.png)
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6 . 如图,在
中,
平分
,
于点
,点
是
的中点.
(1)如图1,
的延长线与
边相交于点
,求证:
;
(2)如图2,线段
、
、
之间满足的数量关系为_________;
【初步运用】
(3)如图3,
中,
平分
,
,垂足为
,过
作
交
于点
,
,
,则
_________;
【灵活运用】
(4)如图4,
中,
,
,点
在
上,
,
,垂足为E,
与
交于点
,线段
、
之间满足的数量关系为_________.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f09785a454d62cbbd2e16ed61ed172a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(1)如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af07faebaf5346f6c3cd2603519d0f60.png)
(2)如图2,线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
【初步运用】
(3)如图3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/650c6c818df102a83ce5159e3208d01a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c0aa4c793921fdaa5d430cb90d78ca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f01c4faacedfe56f5127d6c0cc63cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc11331a7b2d2619b40ee6d34c3bd620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/404728d0ea2699c71e674f30dd984abd.png)
【灵活运用】
(4)如图4,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f81e24376a13d648c2ed0dc73bc710e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3570a95f68349fcd9417fcda62e78e7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbaec9cc53032210714b95af502c7dae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878a63a7899c8226744748a0102eab95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
您最近一年使用:0次
7 . 阅读与思考:
我们知道,如图1,在四边形
中,点
,
,
,
分别是边
,
,
,
的中点,顺次连接
,
,
,
,得到的四边形
是平行四边形.这个平行四边形
是四边形
的中点四边形,也称为瓦里尼翁平行四边形.瓦里尼翁平行四边形与原四边形关系密切.
①当原四边形的对角线满足一定关系时,瓦里尼翁平行四边形可能是菱形、矩形或正方形.
②瓦里尼翁平行四边形的周长与原四边形对角线的长度也有一定关系.
③瓦里尼翁平行四边形的面积等于原四边形面积的一半,此结论可借助图1证明如下:
,分别交
,
于点
,
,
,
分别为
,
中点,
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b39addc1173a458af87ed5c5e3f06466.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64fc74166b877ab80444ded864a737.png)
.
∵
,
分别为
,
中点,
∴________
________(填空1)
∴________
________(填空2)
∴四边形
是瓦里尼翁平行四边形.
任务:
(1)填空1:________
________;填空2:________
________
(2)矩形的瓦里尼翁平行四边形是( )
A.平行四边形 B.菱形 C. 矩形 D.正方形
(3)菱形的瓦里尼翁平行四边形是( )
A.平行四边形 B.菱形 C. 矩形 D.正方形
(4)在图1中,分别连接
,
得到图3,请猜想瓦里尼翁平行四边形
的周长与对角线
,
长度的关系,并证明你的结论.
我们知道,如图1,在四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67d822262ff00915910e5b87d81ad1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
①当原四边形的对角线满足一定关系时,瓦里尼翁平行四边形可能是菱形、矩形或正方形.
②瓦里尼翁平行四边形的周长与原四边形对角线的长度也有一定关系.
③瓦里尼翁平行四边形的面积等于原四边形面积的一半,此结论可借助图1证明如下:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/589786dd7c3a2679c3230b671cd232d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b39addc1173a458af87ed5c5e3f06466.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64fc74166b877ab80444ded864a737.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c93148adbc6e856da9a9d263f485d003.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
∴________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64fc74166b877ab80444ded864a737.png)
∴________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64fc74166b877ab80444ded864a737.png)
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
任务:
(1)填空1:________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64fc74166b877ab80444ded864a737.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad64fc74166b877ab80444ded864a737.png)
(2)矩形的瓦里尼翁平行四边形是( )
A.平行四边形 B.菱形 C. 矩形 D.正方形
(3)菱形的瓦里尼翁平行四边形是( )
A.平行四边形 B.菱形 C. 矩形 D.正方形
(4)在图1中,分别连接
![](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/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
您最近一年使用:0次
8 . 数学活动课上,张老师出示了一个问题:如图1,在
中,
,垂足为E,F为
的中点,连接
,
,求证:
.
①小芳同学由已知条件中点想到了如图2的辅助线.
②小琳同学由已知条件中点想到了如图3的方法.
【类别分析】
(2)小迪同学受此问题启发,将
沿着
(F为
的中点)所在直线折叠,如图4,点C的对应点为点
,连接
并延长交
于点K,在此基础上,小芮同学想:线段
与
之间会有怎样的数量关系呢?请你判断,并证明;
【学以致用】
(3)小怡同学突发奇想,将
沿着过点B的直线折叠,如图5,点A的对应点为点
使
于点H,折痕交
于点M,
交
于点N.小怡提出一个问题:若
的面积为20,
,
,请直接写出
的面积与
的面积的比.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c897a54f2e36bc4b52fba74b41c89d2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/193ea44749f1c64c8723e84a57d15cb9.png)
①小芳同学由已知条件中点想到了如图2的辅助线.
②小琳同学由已知条件中点想到了如图3的方法.
【类别分析】
(2)小迪同学受此问题启发,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/288abe01824f42cfe725509af5aec4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4c6b0869d7248b6c2f964718a67702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48ad8115387158c8dda205d26968ba11.png)
【学以致用】
(3)小怡同学突发奇想,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5128f894d192816321cabcecdff6c170.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42198b8e646ccfbbec323fd364c4d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f08273d339dc5ddbb89aa67bb8205e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/133760237c0ccf2d6a83786925b6d23c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9830cb9f3785e2b262b99bf831ced636.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b352b7a0265d1bda1caa3f06f2b2dc9.png)
您最近一年使用:0次
9 . 我们把依次连接任意四边形各边中点得到的四边形叫做“中点四边形”.如图,在四边形
中,E、F、G、H分别是边
、
、
、
的中点,依次连接各边中点得到“中点四边形”
.
的形状是 ;
(2)求证:矩形的“中点四边形”是菱形.(画图,写出已知、求证和证明)
![](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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67d822262ff00915910e5b87d81ad1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
(2)求证:矩形的“中点四边形”是菱形.(画图,写出已知、求证和证明)
您最近一年使用:0次
10 . 如图,在
中,
和
的角平分线
与
交于点
,且点
恰好在边
上.
为
的中点;
(2)点
为
的中点,连接
,交
于点
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f10681704c4fccfd37664eaf34391dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bab1b173dbc2cf639f0bb023c9718b06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f79d7939c88e9702962e5917cad290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ffb98f1e3c1317c0db403d3af04bdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ffb98f1e3c1317c0db403d3af04bdc.png)
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f79d7939c88e9702962e5917cad290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8120f2eeb724c756b5f84a14c6df527.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc07302fd45bf77fdd9eca7bc8dbea5e.png)
您最近一年使用:0次