阅读下列材料,并完成相应的任务.
托勒密定理:
托勒密(Ptolemy)(公元90年~公元168年),希腊著名的天文学家,他的要著作《天文学大成》被后人称为“伟大的数学书”,托勒密有时把它叫作《数学文集》,托勒密从书中摘出并加以完善,得到了著名的托勒密(Ptolemy)定理.
圆内接四边形中,两条对角线的乘积等于两组对边乘积之和.
已知:如图1,四边形ABCD内接于⊙O,
求证:AB•CD+BC•AD=AC•BD
下面是该结论的证明过程:
证明:如图2,作∠BAE=∠CAD,交BD于点E.
∵![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b0d8fd3d4c39b7560ef6f8156b4a5b4.png)
∴∠ABE=∠ACD
∴△ABE∽△ACD
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b0d8fd3d4c39b7560ef6f8156b4a5b4.png)
∴AB•CD=AC•BE
∵![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b1d6d384c118954c479ebb1fce33999.png)
∴∠ACB=∠ADE(依据1)
∵∠BAE=∠CAD
∴∠BAE+∠EAC=∠CAD+∠EAC
即∠BAC=∠EAD
∴△ABC∽△AED(依据2)
∴AD•BC=AC•ED
∴AB•CD+AD•BC=AC•(BE+ED)
∴AB•CD+AD•BC=AC•BD
任务:(1)上述证明过程中的“依据1”、“依据2”分别是指什么?
(2)当圆内接四边形ABCD是矩形时,托勒密定理就是我们非常熟知的一个定理: .
(请写出)
(3)如图3,四边形ABCD内接于⊙O,AB=3,AD=5,∠BAD=60°,点C为
的中点,求AC的长.
托勒密定理:
托勒密(Ptolemy)(公元90年~公元168年),希腊著名的天文学家,他的要著作《天文学大成》被后人称为“伟大的数学书”,托勒密有时把它叫作《数学文集》,托勒密从书中摘出并加以完善,得到了著名的托勒密(Ptolemy)定理.
圆内接四边形中,两条对角线的乘积等于两组对边乘积之和.
已知:如图1,四边形ABCD内接于⊙O,
求证:AB•CD+BC•AD=AC•BD
下面是该结论的证明过程:
证明:如图2,作∠BAE=∠CAD,交BD于点E.
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b0d8fd3d4c39b7560ef6f8156b4a5b4.png)
∴∠ABE=∠ACD
∴△ABE∽△ACD
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b0d8fd3d4c39b7560ef6f8156b4a5b4.png)
∴AB•CD=AC•BE
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b1d6d384c118954c479ebb1fce33999.png)
∴∠ACB=∠ADE(依据1)
∵∠BAE=∠CAD
∴∠BAE+∠EAC=∠CAD+∠EAC
即∠BAC=∠EAD
∴△ABC∽△AED(依据2)
∴AD•BC=AC•ED
∴AB•CD+AD•BC=AC•(BE+ED)
∴AB•CD+AD•BC=AC•BD
任务:(1)上述证明过程中的“依据1”、“依据2”分别是指什么?
(2)当圆内接四边形ABCD是矩形时,托勒密定理就是我们非常熟知的一个定理: .
(请写出)
(3)如图3,四边形ABCD内接于⊙O,AB=3,AD=5,∠BAD=60°,点C为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bd6ffb78dad3375efa3b08ab518553d.png)
2019·山西大同·二模 查看更多[5]
更新时间:2019-05-22 14:53:41
|
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解答题-证明题
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适中
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【推荐1】如图,AB是
的弦,直线BC与
相切于点B,
,垂足为D,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/6/28/4b5145e1-026f-4995-8890-e77630d60c8b.png?resizew=156)
(1)求证:AB平分
;
(2)点E是
上一动点,且不与点A、B重合,连接
,若
,求
的度数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2022/6/28/4b5145e1-026f-4995-8890-e77630d60c8b.png?resizew=156)
(1)求证:AB平分
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/781ccc2cdcc1683d5a1366ebce65e689.png)
(2)点E是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c8ba4da3a6b049f010c91ccb4f328ee.png)
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【推荐2】如图,在平面直角坐标系中,点A、B是y轴x轴上的两个定点,点M是线段AB的垂直平分线上的一个动点,以点M为圆心,MA长为半径的圆与x轴正半轴、y轴的负半轴分别交于D、C两点,过点O作AB的垂线与CD交于点F .
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/14/a136cc50-35ee-48ac-aa3c-aa9315acfad9.png?resizew=222)
(1)若∠BOE=28°,求∠CDB的度数 ;
(2)求证:F是CD的中点;
(3)若A(0,4),B(
3,0),连接MF,当点M运动时,MF的值是否发生变化,若不变,求出MF的值;若变化,请说明理由.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/14/a136cc50-35ee-48ac-aa3c-aa9315acfad9.png?resizew=222)
(1)若∠BOE=28°,求∠CDB的度数 ;
(2)求证:F是CD的中点;
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解答题-作图题
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【推荐3】在学了“过任意三角形的三个顶点都可以作一个外接圆”之后,张华同学对“过任意四边形的四个顶点能否作一个外接圆?”进行了探究,下面是他的探究过程,请帮他补充完整.
(1)动手实践:张华先画出了
个不同的四边形,如图所示.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/9/6828885e-24ab-4248-8e9c-923dfcb2c7b3.png?resizew=509)
接着,他在所画出的图①,图②中,分别任选三个顶点用尺规各作了一个圆.请仿照张华的作法在图③中任选三个顶点作一个圆(要求尺规作图,保留作图痕迹,不写作法)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/9/3b9317e5-e87c-4f33-b3e7-72202f5dbc0f.png?resizew=536)
(2)观察、发现:
观察所作的图形,你发现:过任意四边形的四个顶点 能作一个外接圆;(选填“一定”或“不一定”)
(3)测量、猜想:
分别测量(1)中
个不同四边形的各个内角,猜想:如果过一个四边形的四个顶点能作一个圆,那么它相对的两个内角之间存在怎样的数量关系?请写出你的猜想.
(4)证明你的猜想.
(1)动手实践:张华先画出了
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/9/6828885e-24ab-4248-8e9c-923dfcb2c7b3.png?resizew=509)
接着,他在所画出的图①,图②中,分别任选三个顶点用尺规各作了一个圆.请仿照张华的作法在图③中任选三个顶点作一个圆(要求尺规作图,保留作图痕迹,不写作法)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/9/3b9317e5-e87c-4f33-b3e7-72202f5dbc0f.png?resizew=536)
(2)观察、发现:
观察所作的图形,你发现:过任意四边形的四个顶点 能作一个外接圆;(选填“一定”或“不一定”)
(3)测量、猜想:
分别测量(1)中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
(4)证明你的猜想.
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【推荐1】如图,
,
是
的两条直径,弦
于点
,
,
的延长线交于点
,连结
.
![](https://img.xkw.com/dksih/QBM/2022/11/7/3104502830530560/3147606745931776/STEM/714c3de5644e4f5ea87f59465070c58d.png?resizew=164)
(1)求证:
;
(2)若
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,求
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(1)求证:
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(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ef520a2657abdf14fa6818c380b596a.png)
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【推荐2】如图,在
中,点D为
边上的一个动点,以
为直径的
交
于点E,过点C作
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于点F.连接
,若
是
的切线.
;
(2)若
,求直径
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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(2)若
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【推荐1】如图,在等腰三角形
中,
,
,点E为腰
中点,点F在底边
上,且
,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
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名校
【推荐2】如图,在
中,
是直径,点
是弧
的中点,
交
于点
,
是
的中点,
,
,
.
(1)求证:
是
的切线;
(2)求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a7b5adfcac0f46a4cd19da4ebb4a2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf7679c8b4b1e442ce4286d4b0e9c32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04725e51b4870658f74de79403e3898f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/17/0a33f2a3-2496-437f-8c46-dfa0372d70a0.png?resizew=134)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(2)求
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解答题-问答题
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适中
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【推荐1】如图,在梯形
中,
,
,
,
,
.
![](https://img.xkw.com/dksih/QBM/2020/5/23/2468805769084928/2470106704142337/STEM/ce7cb0d4-00c4-46b2-8cb7-e079aec63bd0.png?resizew=147)
(1)求线段
的长;
(2)联结
,交对角线
于点
,求
的余切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e0a957a55460c72673c0f2ee90dbb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7788830ed1cb3b9c5988f70f43595f2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f81e24376a13d648c2ed0dc73bc710e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0effa659739bfdc9e395c72fa38ffbe.png)
![](https://img.xkw.com/dksih/QBM/2020/5/23/2468805769084928/2470106704142337/STEM/ce7cb0d4-00c4-46b2-8cb7-e079aec63bd0.png?resizew=147)
(1)求线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
(2)联结
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/565d0de7845e77737b774e8db6d43ee1.png)
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【推荐2】如图,AB为⊙O直径,CD为⊙O的一条弦,AB与CD交于点M,点E在AD的延长线上,且∠BED=∠ACD.
![](https://img.xkw.com/dksih/QBM/2022/2/23/2922866887393280/3030832042647552/STEM/a0430f11bf2e43e09fbd7f9cbfcd74e1.png?resizew=140)
(1)判断BE与⊙O的位置关系,并说明理由;
(2)若CD∥BE,AC=4
,AM=CD,求BD的长.
![](https://img.xkw.com/dksih/QBM/2022/2/23/2922866887393280/3030832042647552/STEM/a0430f11bf2e43e09fbd7f9cbfcd74e1.png?resizew=140)
(1)判断BE与⊙O的位置关系,并说明理由;
(2)若CD∥BE,AC=4
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
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