1 . 《几何原本》卷2的几何代数法
几何方法研究代数问题
成了后世西方数学家处理问题的重要依据.通过这一原理,很多的代数的公理或定理都能够通过图形实现证明,也称之为无字证明;如图所示图形,点D、F在圆O上,点C在直径AB上,且
,
,
于点E,设
,
,该图形完成
的无字证明.
图中线段__________ 的长度表示![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa6888f782bb98e9c86825597000bddd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd995178601c2ad7b40f973d268c7bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee95969dd8c5d5e0a65d579d0d14b200.png)
图中线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa6888f782bb98e9c86825597000bddd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/20/919e8727-7bba-41c2-b8d0-c4802f9abc86.png?resizew=162)
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2 . 阿波罗尼斯(约公元前262-190年)证明过这样一个命题:平面内到两定点距离之比为常数
且
的点的轨迹是圆,后人将这个圆称为阿波罗尼斯圆.在棱长为6的正方体
中,点
是BC的中点,点
是正方体表面
上一动点(包括边界),且满足
,则三棱锥
体积的最大值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84ec442066a23584bfb5699da59e85f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15bb8775b827a649b07b6c2f8c3ea284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4f2d2ef6661d1808fed0cbd1b0fa53d.png)
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名校
3 . 古希腊数学家欧几里得所著《几何原本》中的“几何代数法”,很多代数公理、定理都能够通过图形实现证明,并称之为“无字证明“如图,
为线段
中点,
为
上的一点.以
为直径作半圆,过点
作
的垂线,交半圆于
.连结
,
,
,过点
作
的垂线,垂足为
.设
,
,则图中线段
,线段
,线段________
;由该图形可以得出
,
,
的大小关系为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/16/3143184d-bc71-40e6-a813-5901a2a2c546.png?resizew=210)
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6卷引用:辽宁省丹东市2020-2021学年高一上学期期末数学试题
4 . 正多面体也称柏拉图立体,被誉为最有规律的立体结构,是所有面都只由一种正多边形构成的多面体(各面都是全等的正多边形).数学家已经证明世界上只存在五种柏拉图立体,即正四面体、正六面体、正八面体、正十二面体、正二十面体.已知一个正八面体ABCDEF的棱长都是2(如图),P,Q分别为棱AB,AD的中点,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3350049b484df2df02602524fa047c6.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3350049b484df2df02602524fa047c6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/21/b5ade79c-80e9-4575-aa6a-f05148f91559.png?resizew=145)
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10卷引用:河南省创新联盟2022-2023学年高二上学期第一次联考(B卷)数学试题
河南省创新联盟2022-2023学年高二上学期第一次联考(B卷)数学试题湖北省襄阳市部分学校2022-2023学年高二上学期9月联考数学试题湖北省孝感市部分学校2022-2023学年高二上学期9月联考数学试题(已下线)第04讲 空间向量在立体几何中的应用(练,理科专用)湖北省襄阳市第二中学2022-2023学年高二上学期9月月考数学试题辽宁省沈阳市第十中学2022-2023学年高二上学期10月月考数学试题(已下线)专题16 空间向量及其应用(练习)-1(已下线)模块二 专题1 《空间向量与立体几何》单元检测篇 B提升卷(苏教 )辽宁省沈阳市第十中学2022-2023学年高二上学期第一阶段考试数学试题(已下线)1.2 空间向量基本定理【第三练】
名校
5 . 《几何原本》中的几何代数法(用几何方法研究代数问题)成了后世西方数学家处理问题的重要依据,通过这一方法,很多代数公理、定理都能够通过图形实现证明,并称之为“无字证明”.设
,
,称
为
,
的调和平均数.如图,
为线段
上的点,且
,
,
为
中点,以
为直径作半圆.过点
作
的垂线,交半圆于
,连结
,
,
.过点
作
的垂线,垂足为
.则图中线段
的长度是
,
的算术平均数
,线段
的长度是
,
的几何平均数
,线段__ 的长度是
,
的调和平均数
,该图形可以完美证明三者的大小关系为__ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eafd0e253a0a62512d50c656de3dc2e9.png)
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解题方法
6 . 三角形面积公式
(1)三角形的面积等于两边及两边夹角的正弦值之积的一半,即![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ec636d85f67b42dafd7d78dfcf3f1f9.png)
______ =______ .
证明:建立如图所示的平面直角坐标系,设
,则有
所以
.同理,
的面积还可以表示为
和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4cafc1c7cb266cee6688897751c358f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/23/3ff6246c-fd69-419d-8095-8f22a1af9298.png?resizew=205)
(2)
(请用正弦定理自行证明).
(1)三角形的面积等于两边及两边夹角的正弦值之积的一半,即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ec636d85f67b42dafd7d78dfcf3f1f9.png)
证明:建立如图所示的平面直角坐标系,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c62ede314dfcdeb672a6c3283ba1644.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0352186606719dd9bb81edf7ef14365f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0baa8e9985597168988f8087c4cdbf9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65e764ca37a5ec48b009f16bbd386c06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4cafc1c7cb266cee6688897751c358f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/23/3ff6246c-fd69-419d-8095-8f22a1af9298.png?resizew=205)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d530f66c8c0ca63c6d7f2192fbce352e.png)
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2022高三·全国·专题练习
7 . 阿波罗尼斯(约公元前
年)证明过这样一个命题:平面内到两定点距离之比为常数
的点的轨迹是圆,后人将这个圆称为阿氏圆,已知
、
分别是圆
,圆
上的动点,
是坐标原点,则
的最小值是 __ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d671b358f4e5a8062e5b97f5c0555aa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be776ec968ef225c21c030a535621bf.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb086b6a35662c39159d1228bdd11f1.png)
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解题方法
8 . 数学家Dandelin用来证明一个平面截圆柱得到的截口曲线是椭圆的模型(称为“Dandelin双球”).如图,在圆柱内放两个大小相同的小球
,使得两球球面分别与圆柱侧面相切于以
为直径且平行于圆柱底面的圆
和
,两球球面与斜截面分别相切于点
,点
为斜截面边缘上的动点,则这个斜截面是椭圆.若图中球的半径为3,球心距离
,则所得椭圆的离心率是___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d096cd7bd8a5a2219fd7dd166bbb8460.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3956369191d0383bdb8786134da7dd9f.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/647af54e46e22ea0160071ca6eacb1a5.png)
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2022-11-16更新
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763次组卷
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4卷引用:江西省名校联盟2022-2023学年高二上学期期中联考数学试题
9 . 我国汉代数学家赵爽为了证明勾股定理,创制了一幅“勾股圆方图”,后人称其为“赵爽弦图”.类比赵爽弦图,由3个全等的小三角形拼成如图所示的等边
,若
的边长为
且
,则
的面积为_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
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10 . 利用数学归纳法证明“
”时,由
到
时,左边应添加因式______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7d488bc552b77155b56464011ec9452.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b00f4eb7f1bd2ccefbabf0c1dfa8f69.png)
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