解题方法
1 . 对任意两个非零向量
,定义
.若非零向量
,满足
,向量
与
的夹角是锐角,且
是整数,则
的取值范围是_____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662917aedec92809a13618093c8e0c3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63c7ac4b4b9924ef29792fa6ac198f06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8e8b95a61af300412fc65f846089028.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26739971f3cae6d4790870c344b5cf89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b35bee611ba4204db25848d5925c618d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4db35dbee504e3e66bfd03c24e4b7322.png)
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解题方法
2 . 已知平面向量
,
,且
,
,向量
满足
,则
取最小值时,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5085e3cdef9ea6c564e079f745d6fdb.png)
_________________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9a5451e35042092a0cdc5971a4d55e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e37157acb52896b1d3d067adc7d381b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d366d8fbb7258ee051f49977441e14a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59d3e5619718e1bb2592caf48bcc2718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5b0a4509691f41732a190fceef0175.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5085e3cdef9ea6c564e079f745d6fdb.png)
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解题方法
3 . 中国文化中的太极八卦图蕴含了现代哲学中的矛盾对立统一规律,如图1是八卦模型图,其平面图形记为图2中的正八边形
,其中
,若点P是其内部任意一点,则
的取值范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17d4a6cf11cda87b3dfafaecdec683f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52705567101a48893de582656ef41527.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85b8d017e422f158fe09ed9c867ecfe7.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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4 . 数学中有很多相似的问题,
材料一:十七世纪法国数学家,被誉为业余数学家之王的皮埃尔·德·费马提出了一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小”,他的答案是:“当三角形的三个内角均小于
时,所求的点为三角形的正等角中心,即该点与三角形的三个顶点的连线两两成角
,当三角形有一内角大于或等于
时,所求点为三角形最大内角的顶点”,在费马问题中所求的点称为费马点.
材料二:布洛卡点,也叫“勃罗卡点”,定义为:已知
内一点
满足
,则称
为
的布洛卡点,
为
的布洛卡角,1875年,三角形的这一特殊点,被一个数学爱好者——法国军官布洛卡重新发现,并用他的名字命名.
已知
,
,
分别是
的内角
,
,
的对边,且
.
(1)求
;
(2)若
为
的费马点,且
,求
的值;
(3)若
为锐角三角形,
为
的布洛卡点,
为
的布洛卡角,证明:
.
材料一:十七世纪法国数学家,被誉为业余数学家之王的皮埃尔·德·费马提出了一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小”,他的答案是:“当三角形的三个内角均小于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00e4979100d4078609e253e2f99eed0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00e4979100d4078609e253e2f99eed0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00e4979100d4078609e253e2f99eed0b.png)
材料二:布洛卡点,也叫“勃罗卡点”,定义为:已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c25d734ea37934683320c146c2c67a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
已知
![](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/e428e7a09732be85c1224e9c8f6a71c5.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/481b91aa00df0bf153f717d87d1b12f7.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54728823efd2745d64ae9921f8807917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1424f6ac5e01f56e2d486c68a5be1a0.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26f61d98c51b9f0344cf7b4562680f45.png)
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解题方法
5 . 如图,在平面四边形
中,
,
,记
与
的面积分别为
, 则
的值为_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27db558e8db4c957654c8e5cecd2d2dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b899bfd0597444039b66657878d37533.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a783e5ffcf7a4ea9e531ea76199487.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76b982b4ed3b75609bda6af3914986d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16ae89bd2dd0a18e1afb5b1a1abd0efd.png)
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解题方法
6 . 已知圆C:
,直线l:
,若l与圆C交于A,B两点,设坐标原点为O,则
的最大值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ca01d1c400707e011745ad16199da13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ef6d00ddabcb1bb41fae534f5183ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df09bff72dab3e6555a0597b638a3bd0.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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7日内更新
|
48次组卷
|
2卷引用:2024届广西壮族自治区贵港市高考模拟预测数学试题
2024高三·全国·专题练习
解题方法
7 . “费马点”是由十七世纪法国数学家费马提出并征解的一个问题.该问题是:“在一个三角形内求作一点,使其与此三角形的三个顶点的距离之和最小.”意大利数学家托里拆利给出了解答,当
的三个内角均小于
时,使得
的点
即为费马点;当
有一个内角大于或等于
时,最大内角的顶点为费马点.在
中,内角
,
,
的对边分别为
,
,
.
(1)若
.
①求
;
②若
的面积为
,设点
为
的费马点,求
的取值范围;
(2)若
内一点
满足
,且
平分
,试问是否存在常实数
,使得
,若存在,求出常数
;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1eab88a16df610f20dd46a44ba098d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](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)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b7f7180b86108862c7aa44c950f872a.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab7aaa871ceb78e5b80b531a7cf4f1c9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec15e5cb6d4dc2cf6ba0bedd87514448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca347a0ea5e4d813a81407796be5fea7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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2024高三·全国·专题练习
解题方法
8 . 费马点是在三角形中到三个顶点距离之和最小的点.具体位置取决于三角形的形状,如果三角形的三个内角均小于
,费马点是三角形内部对三边张角均为
的点;如果三角形有一个内角大于或等于
,费马点就是该内角所在的顶点.已知△ABC中,角A,B,C所对的边分别为a,b,c,O为费马点.
(1)若
,
,
,求
的值;
(2)若
,
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/783d6adfa8fb1352679c5185258d842a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a3c442579603164f3fc19458677d307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb5bac75f36bb1dc5c8190d4dbe681d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/569b5df7e2e4642091364efefe8dddf9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa1466856bf2570685d3629c1f813748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c2bd0a04afc05de0f6a86ada42411f2.png)
您最近一年使用:0次
2024高三·全国·专题练习
9 . 十七世纪法国数学家、被誉为业余数学家之王的皮埃尔·德·费马提出的一个著名的几何问题:“已知一个三角形,求作一点,使其与这个三角形的三个顶点的距离之和最小.”它的答案是:“当三角形的三个角均小于
时,所求的点为三角形的正等角中心,即该点与三角形的三个顶点的连线两两成角
;当三角形有一内角大于或等于
时,所求点为三角形最大内角的顶点.”在费马问题中所求的点称为费马点. 试用以上知识解决下面问题:已知
的内角
所对的边分别为
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40ec9cff8627e76b61e6474e57d7a7ef.png)
(1)求
;
(2)若
,设点
为
的费马点,求
;
(3)设点
为
的费马点,
,求实数
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40ec9cff8627e76b61e6474e57d7a7ef.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44ac38c5cc951497a4a37778b191bcce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b8f8a1e38db0e55b9b1934569b24e74.png)
(3)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b01862dfc85d45102a1343c36cb6dfe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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解题方法
10 . 如图,已知梯形
中,
,
,点
,
分别为线段
,
上的动点,
,点
为线段
中点,则以下说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f79863ffcfa63117ca6741b20a48e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bc9cd61f54d45b905a295b35cdbe9d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7031367b886986db7850cdc47e2bec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
A.若![]() ![]() | B.![]() |
C.![]() | D.若![]() ![]() ![]() |
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