1 . 我国汉代数学家赵爽为了证明勾股定理,创造了一幅“勾股圆方图”,后人称其为“赵爽弦图”.类比赵爽弦图,用3个全等的小三角形拼成了如图所示的等边
,若
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a71a9d21f77e9535de152bb33f802bb.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/221a091e823526ce02a78be01068c01d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c0ba1776a7c0bac5141407836e12153.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a71a9d21f77e9535de152bb33f802bb.png)
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2024-06-13更新
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434次组卷
|
2卷引用:四川成华区某校2023-2024学年高一下学期期中考试数学试题
23-24高二下·全国·课前预习
2 . 等差中项
(1)条件:如果
成等差数列.
(2)结论:那么
叫做
与
的等差中项.
(3)满足的关系式是________
温警提醒(1)任意两个实数都有等差中项.
(2)应用等差中项法也可证明一个数列为等差数列,即
为等差数列.
(1)条件:如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf6726a4207c053c937cf221120dea1.png)
(2)结论:那么
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
(3)满足的关系式是
温警提醒(1)任意两个实数都有等差中项.
(2)应用等差中项法也可证明一个数列为等差数列,即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aa511f5869c3ac911876fc9af0f51b1.png)
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名校
3 . 《几何原本》卷2的几何代数法(以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.现有如图所示的图形,点
在以
为直径的半圆上,
为圆心,点
在半径
上(不与
点重合),且
.设
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bc69e886c7afedf1c9233e9a2a6870.png)
__________ (用
表示),由
可以得出的关于
的不等式为__________ .
![](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/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ebef5bab02280cdc99cc7f689135cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294726f8e596ce099d050ebcd538e421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bc69e886c7afedf1c9233e9a2a6870.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76edbc800f52f6f8f710b1d7179fb31f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/26/82b41fab-4a23-4fdb-8191-a5a97d6b0134.png?resizew=160)
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4 . 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
.连接
,
,
,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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3d296e0d7154a170cb7d3ae42989b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bcccda6e75578c160552bcb1d7f160b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9ce2f12cd473b0877cb01872ec45141.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8a24490af6cdebc539613da0a98d762.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26234bb9c659eb48da0247dd6a465d65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/15/3757da65-71fc-4c20-9430-975b3469b269.png?resizew=185)
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解题方法
5 . 《几何原本》中的几何代数法是以几何方法研究代数问题,这种方法是后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公理或定理都能够通过图形实现证明,也称之为无字证明,现有图形如图所示,C为线段
上的点,且
,
,O为
的中点,以
为直径作半圆,过点C作
的垂线交半圆于D,连接
,
,
,过点C作
的垂线,垂足为E,若不添加辅助线,则该图形可以完成的所有无字证明为__________ .(填写序号)
①
;②
;
③
;④
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3d296e0d7154a170cb7d3ae42989b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a88b719166fcc1431f876bc8c5656c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/1/3829f1ef-4d36-4411-b991-19beda49ad27.png?resizew=169)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb90c316d8a99694396de80ed0b0cf25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09adf1ac3103c5ddfe7a92856cb2afbe.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf323321cd5f80e909c243941c33f7db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a305a51783a797bdda25197e090feb05.png)
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6 . 科拉茨是德国数学家,他在1937年提出了一个著名的猜想:任给一个正整数
,如果
是偶数,就将它减半(即
);如果
是奇数,则将它乘3加1(即
),不断重复这样的运算,经过有限步后,一定可以得到1.这是一个很有趣的猜想,但目前还没有证明或否定.如果对正整数
(首项)按照上述规则施行变换后得到
,依次施行变换后所得到的数组成数列
,
是数列
的前
项和,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b024dcba89b9bc12300583e25c1ed90.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bab29cb6e1d21628f312a23f76f44d5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0e19f7bfb0ee59fc93e6e822a0658af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54b71ef6cb9c5d494692d40a9ef279f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b024dcba89b9bc12300583e25c1ed90.png)
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2023-11-22更新
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285次组卷
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3卷引用:广西壮族自治区广西贵港市、百色市、河池市2023-2024学年高三上学期11月质量调研联考数学试题
广西壮族自治区广西贵港市、百色市、河池市2023-2024学年高三上学期11月质量调研联考数学试题广西贵港市、百色市、河池市2024届高三上学期11月质量调研联考数学试题(已下线)考点16 几类特殊的数列模型 2024届高考数学考点总动员【练】
名校
解题方法
7 . 我国后汉时期的数学家赵爽通过弦图利用出入相补法证明了勾股定理,在我国历史上还有多人通过出入相补法证明过勾股定理,如下图为我国清末数学家华蘅芳证明勾股定理时构造的图形,在该图中
是以
为斜边的直角三角形,分别以
为边作3个正方形,点
在直线
上,
,记
的周长与面积分别为
,则
的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0e08a39c6619123557148d195abfbe.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/ad9c4e14fcf121dca94b9a5549a93288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3de8fa2fa9271308932731a418f21f95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f484be7264a86c5dcfe4e25d6459ac43.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/23/670c1694-0cdb-4130-9fd8-cc9450265163.png?resizew=168)
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名校
8 . 科拉茨是德国数学家,他在1937年提出了一个著名的猜想:任给一个正整数
,如果
是偶数,就将它减半(即
);如果
是奇数,则将它乘3加1(即
),不断重复这样的运算,经过有限步后,一定可以得到1.这是一个很有趣的猜想,但目前还没有证明或否定.如果对正整数
(首项)按照上述规则施行变换后的第8项为1(注:1可以多次出现),则满足条件的
的所有不同值的和为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bab29cb6e1d21628f312a23f76f44d5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0e19f7bfb0ee59fc93e6e822a0658af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2023-04-03更新
|
2292次组卷
|
6卷引用:湖南师范大学附属中学2023届高三一模数学试题
名校
解题方法
9 . 无字证明(proof without words)是指仅用图象而无需文字解释就能不证自明的数学命题,如图是某三角恒等式的无字证明,那么该图证明的三角恒等式为__________ .
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/14/13cf9ace-987e-45b3-9f16-a17dbcc52b23.png?resizew=435)
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2023-06-13更新
|
604次组卷
|
2卷引用:湖南师范大学附属中学2022-2023学年高一下学期第二次大练习数学试题
10 . 《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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ebef5bab02280cdc99cc7f689135cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b757f0c42ae5c9a2d6a4b19e5877b27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/378eaf7b1f0e24833bc1c762935b04d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3d296e0d7154a170cb7d3ae42989b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51161a40f54f4791ff3512a34f6242cb.png)
![](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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