1 . 《几何原本》是古希腊数学家欧几里得得所著的一部数学著作,在《几何原本》第六卷给出了内角平分线定理,其内容为:在一个三角形中,三角形一个内角的角平分线内分对边所成的两条线段,与这个角的两邻边对应成比例.例如,在
中(图1),
为
的平分线,则有
.
(2)如图2,已知
的重心为
,内心为
,若
的连线
.求证:
.
![](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/608bf0cfbbe809837adec2755fcd2901.png)
(2)如图2,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57b8fc74eea80b1ccf11d16ad7b3178a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/981b01ddc1aa5fcf155ad41307d22b17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a94a70686cb9c91ec9705bed47dc663.png)
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2 . 对于平面向量
,定义“
变换”:
,其中
表示
中较大的一个数,
表示
中较小的一个数.若
,则
.记
.
(1)若
,求
及
;
(2)已知
,将
经过
次
变换后,
最小,求
的最小值;
(3)证明:对任意
,经过若干次
变换后,必存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4fde5542ad04744c14f912648f3aa0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41bd66e602e9c043218806708e943c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb00071815c94c090a4095b4964fefb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7bc9573b3a8758511c63731db18183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96340894e8fb63c00d778b4d654d0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f7bc9573b3a8758511c63731db18183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701ab98a2bf1135cd989822b0738e11d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/484c1b7bc2fc5677406e20180f667200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0624499e16b73afec432dd1afd6153d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b162d1d5bfaa7760678ea3d624beb171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5c19921380da55f5f1a00809a34503.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35234a3829d238ea479fef9cec166468.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8aceb3666a9d49ef40c39eac116ccd5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20cfb8c8707c3960bf1fd46b805e481d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a887552671e6d4df390320ee9a36150.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)证明:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f389ec068eb1d1aa586b79097d70a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dcd78ec8777a8e6e5b32222cdb15c05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06296b9023c1dca6f44b8297842bef7c.png)
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名校
3 . 平面几何中有一定理如下:三角形任意一个顶点到其垂心(三角形三条高所在直线的交点)的距离等于外心(外接圆圆心)到该顶点对边距离的2倍.已知
的垂心为D,外心为E,D和E关于原点O对称,
.
(1)若
,点B在第二象限,直线
轴,求点B的坐标;
(2)若A,D,E三点共线,椭圆T:
与
内切,证明:D,E为椭圆T的两个焦点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24e12c97516329a6776fe48c450d93b.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25c8ef6f3640bd70e40f3b591c8bcc14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05b45db8dd8768994af51206565379fd.png)
(2)若A,D,E三点共线,椭圆T:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2024-05-08更新
|
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5卷引用:广西钦州市2024届高三年级第三次教学质量监测 数学
名校
解题方法
4 . 已知函数
的图象在定义域
上连续不断.若存在常数
,使得对于任意的
,
恒成立,称函数
满足性质
.
(1)若
满足性质
,且
,求
的值;
(2)若
,试说明至少存在两个不等的正数
,同时使得函数
满足性质
和
.(参考数据:
)
(3)若函数
满足性质
,求证:函数
存在零点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/870ebc2f7aabb028024894568d749934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/094f977194228bed828f3507f5898934.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2a2c48c3896c9f07bc82434e30020fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c1ff5cb5a9d88ed7db2c06683c3e355.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bea0dd7e474bcd04db2544427ba0488.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/196be101149acfb6a6c4ceca7fc96828.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0feacb36911be3ca27b87449754b28d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/842d905700b5635303a740bd0109ff0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f5dd698ddbe275267809650dc551e34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad9b41127e7230a15dcdc5cae08739c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/879f7ee2372a171567ae512f66216d38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f3ab85db456b851bb7bed23fc9a187f.png)
(3)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c1ff5cb5a9d88ed7db2c06683c3e355.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
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2021-12-15更新
|
769次组卷
|
8卷引用:广西钦州市2022-2023学年高一上学期期末教学质量监测数学试题
广西钦州市2022-2023学年高一上学期期末教学质量监测数学试题北京市海淀区2019-2020学年高一上学期期末调研数学试题(已下线)第8章 函数应用 单元综合检测(难点)(单元培优)-2021-2022学年高一数学课后培优练(苏教版2019必修第一册)广东省茂名高州市2021-2022学年高一上学期期末数学试题福建省莆田第一中学2021-2022学年高一下学期期初学科素养能力竞赛数学试题北京市海淀实验中学2021-2022学年高一下学期期中数学试题江西省宜春市宜丰县宜丰中学2022-2023学年高一下学期期末考试数学试题北京市日坛中学2023-2024学年高一上学期期中考试数学试题