1 . 已知正三角形
的边长是
,若
是
内任意一点,那么
到三角形三边的距离之和是定值
.这是平面几何中一个命题,其证明常采用“面积法”.如图,设
到三边的距离分别是
、
、
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bec1bece500c180231015dc713bd1935.png)
,
为正三角形
的高
,即
.运用类比法猜想,对于空间正四面体,存在什么类似结论,并用“体积法”证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14af83ff12c1d6d35d5335827402fba7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cad4595d5352b2884568a59d8d766a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bec1bece500c180231015dc713bd1935.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66ed01b3eb15d5253f1ab94150dd1582.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14af83ff12c1d6d35d5335827402fba7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0f691b1b21694bf9a45041a00516222.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/596c3623-3e97-4114-b65b-f6eb0faccbc2.png?resizew=152)
您最近一年使用:0次
2 . 如图1,已知
中,
,点
在斜边
上的射影为点
.
![](https://img.xkw.com/dksih/QBM/2018/7/8/1984043695022080/1985696665255936/STEM/5f681ef49f6b4dccac078de8f9fda743.png?resizew=400)
(Ⅰ)求证:
;
(Ⅱ)如图2,已知三棱锥
中,侧棱
,
,
两两互相垂直,点
在底面
内的射影为点
.类比(Ⅰ)中的结论,猜想三棱锥
中
与
,
,
的关系,并证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53502463cc76201000e02df314e58769.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83640592853a53872d7af69c0cffc1bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://img.xkw.com/dksih/QBM/2018/7/8/1984043695022080/1985696665255936/STEM/5f681ef49f6b4dccac078de8f9fda743.png?resizew=400)
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/520306e216b3af1fe08c1762e200610f.png)
(Ⅱ)如图2,已知三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.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/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d58f9019097bd05037aefd5c322916.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/48f3c9abbd78e9a6840ee5f30381daac.png)
您最近一年使用:0次
2018-07-10更新
|
540次组卷
|
5卷引用:河北省衡水市第十三中学2019届高三质检(四)文科数学试题
河北省衡水市第十三中学2019届高三质检(四)文科数学试题【全国市级联考】安徽省蚌埠市2017-2018学年高二下学期期末考试数学(理)试题【全国市级联考】安徽省蚌埠市2017-2018学年高二下学期期末考试数学(文)试题【全国百强校】湖南省湘潭县一中、双峰一中、邵东一中、永州四中2018-2019学年高二下学期优生联考数学试题(已下线)2.1.1 合情推理-2020-2021学年高二数学(理)课时同步练(人教A版选修2-2)
3 . 先阅读下列题目的证法,再解决后面的问题.
已知
,且
,求证:
.
证明:构造函数
,
则
,
因为对一切
,恒有
,
所以
,
从而得
.
(1)若
,请由上述结论写出关于
的推广式;
(2)参考上述证法,请对你推广的结论加以证明.
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2902a0c1309fe2d5490d6753d98d777.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53444e75341e1f40bf7d02c9ae6c47bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7397d7eab4d6f27a9bf0444c1b5ea889.png)
证明:构造函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bde9e414278cb72701ae87ba38bdab8.png)
则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bec845e92f934ce8c76dca49c872820.png)
因为对一切
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32a6e5b8aaed692d8be521e82df5a230.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e53c18d3b7ba956a1ec73418e6db9731.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/696b9008f29731ef9934398429e5b75f.png)
从而得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7397d7eab4d6f27a9bf0444c1b5ea889.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1b2d15f49b77537ea0efbd8b7fb6cc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09fc545b119b0b1c9e496e1c6ff9c25e.png)
(2)参考上述证法,请对你推广的结论加以证明.
您最近一年使用:0次
2018-06-24更新
|
247次组卷
|
13卷引用:河北省枣强中学2016-2017学年高二下学期期末考试数学(理)试题
河北省枣强中学2016-2017学年高二下学期期末考试数学(理)试题(已下线)2013-2014学年湘教版高二数学选修2-2基础达标6.1练习卷2015-2016学年安徽省六安一中高二下第一次段考文数学卷2016-2017学年江西省新余市高二上学期期末考试文数试卷高中数学人教A版选修2-2 综合复习与测试 (4)陕西省澄城县2017-2018学年高二下学期期中考试数学(理)试题黑龙江省海林市朝鲜族中学人教版高中数学选修1-2同步练习:第二章 推理与证明单元测评广东省佛山市第三中学2018-2019学年第二学期第一次段考高二理科数学试题上海市浦东新区川沙中学2015-2016学年高一上学期期中数学试题安徽省马鞍山二中2018-2019学年高二下学期期中文科数学试题沪教版(2020) 必修第一册 达标检测 第二章 章测试河南省南阳市第一中学2021-2022学年高二下学期第二次月考文科数学试题河南省郑州市第十九高级中学2020-2021学年高二下学期3月月考理科数学试题
4 .
(1)在
中,内角
,
,
的对边分别为
,
,
,且
,证明:
;
(2)已知结论:在直角三角形中,若两直角边长分别为
,
,斜边长为
,则斜边上的高
.若把该结论推广到空间:在侧棱互相垂直的四面体
中,若三个侧面的面积分别为
,
,
,底面面积为
,则该四面体的高
与
,
,
,
之间的关系是什么?(用
,
,
,
表示
)
(1)在
![](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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96de116c48275fb4810136abc57a286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26c3cefa3cb1c160b392850ed1558a7.png)
(2)已知结论:在直角三角形中,若两直角边长分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2114a1af34caa683f072770325f9781.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
您最近一年使用:0次
2018-06-01更新
|
445次组卷
|
2卷引用:【全国市级联考】河北省邢台市2017-2018学年高二下学期第三次月考数学(文)试题
5 . (1)已知:
,
,证明:
;
(2)已知
,证明:
,并类比上面的结论,写出推广后的一般性结论(不需证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc93e18ce083103103113d6d6d845618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b40b1544e62be8b9e9f4dc9f2c0c74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e0989f71c74fdfd6f97d8c9f481ba1.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6fea464f1b120d173d5070b2899bac4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5bb8a8776010a3eb1f502dbc28537f6.png)
您最近一年使用:0次
2016-12-04更新
|
513次组卷
|
3卷引用:2015-2016学年河北武邑中学高二下5.8周考文科数学卷