1 . 如图,AB为⊙O的直径,BD是⊙O的切线,连接AD交⊙O于E,若BD∥CE,AB交CE于M,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e4b0311cb57d78f465a57dc7f26353c.png)
![](https://img.xkw.com/dksih/QBM/2018/12/5/2090404214185984/2094345211813888/STEM/647c1bdc-8188-4995-bad2-424c20db0e6d.png)
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2 . 已知PQ与圆O相切于点A,直线PBC交圆于B、C两点,D是圆上一点,且AB∥DC ,DC的延长线交PQ于点Q.
![](https://img.xkw.com/dksih/QBM/2016/4/12/1572588562546688/1572588568035328/STEM/ac6ae66cfecc4ccab7864831b5a9c67e.png)
(1)求证: AC2=CQ×AB;
(2)若AQ=2AP,AB=
,BP=2,求QD.
![](https://img.xkw.com/dksih/QBM/2016/4/12/1572588562546688/1572588568035328/STEM/ac6ae66cfecc4ccab7864831b5a9c67e.png)
(1)求证: AC2=CQ×AB;
(2)若AQ=2AP,AB=
![](https://img.xkw.com/dksih/QBM/2016/4/12/1572588562546688/1572588568035328/STEM/6c62ee9cadb44de8820f95ed21ba65d8.png)
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3 . (选修4-1:几何证明选讲)在圆O中,AB,CD是互相平行的两条弦,直线AE与圆O相切于点A,且与CD的延长线交于点E,求证:AD2=AB·ED.
![](https://img.xkw.com/dksih/QBM/2017/6/1/1699409391165440/1699652214628352/STEM/9743b396f71848a6950fd9e471ee5b55.png?resizew=239)
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4 . 选修4—1:几何证明选讲
如图,
内接于圆
,
平分
交圆
于点
,过点
作圆
的切线交直线
于点
.
![](https://img.xkw.com/dksih/QBM/2015/4/15/1572073022324736/1572073028354048/STEM/8b152a1213744b2fa0813ae3d7523138.png)
(1)求证:
;
(2)求证:
.
如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2333da32638ec636fa1e69e750ca4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6062582a0a1679aa777451f25008411d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://img.xkw.com/dksih/QBM/2015/4/15/1572073022324736/1572073028354048/STEM/8b152a1213744b2fa0813ae3d7523138.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96db6ed42a67239083c6b36475147e3c.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5777d129b34192134305e1c61e60876.png)
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12-13高三上·辽宁本溪·期末
5 . 如图,D,E分别是AB,AC边上的点,且不与顶点重合,已知
,
为方程
的两根,
(1) 证明
四点共圆;
(2) 若
,求
四点所在圆的半径.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b662797f6678e2475f4631563694e37a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea77ba313fcc751481ac1ca214df3fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6ed516191ef7d1805cd7cf945232d79.png)
(1) 证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b226772d996dafd09c37b5bad17d54fd.png)
(2) 若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96f402d5cb7a7d2ea738391f39a00897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b226772d996dafd09c37b5bad17d54fd.png)
![](https://img.xkw.com/dksih/QBM/2012/2/15/1570738702614528/1570738707406848/STEM/bd53a0f954f849d9b5e078e4efc9698f.png?resizew=150)
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2011·山西·一模
6 . 选做题(10分.请考生必须在22、23题中任选一题做答,如果多做,则按所做的第一题记分)
22.
选修4-1:几何证明选讲
在△ABC中,AB=AC,过点A的直线与其外接圆交于点P,交BC延长线于点D.
(Ⅰ)求证:
;
(Ⅱ)若AC=2,求AP•AD的值.
22.
选修4-1:几何证明选讲
在△ABC中,AB=AC,过点A的直线与其外接圆交于点P,交BC延长线于点D.
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f44caa95b907ab06f85da6f52b9213.png)
(Ⅱ)若AC=2,求AP•AD的值.
![](https://img.xkw.com/dksih/QBM/2011/3/9/1570032299630592/1570032305381376/STEM/fd92e888cb9043e0bcd8d3422ae0a244.png?resizew=216)
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2012·海南省直辖县级单位·一模
7 . 如图设
为线段
中点,
与
交于点
,且
交
于
交
于
.
![](https://img.xkw.com/dksih/QBM/2012/3/20/1570811241472000/1570811247280128/STEM/3bb7730a3928426e89459eab1ee1f473.png?resizew=194)
(1)写出图中三对相似三角形,并对其中一对作出证明;
(2)连结
,设
,求
长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2be36353f9b1353cb9a38bc26e814521.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a424b50eaeafa6f302ffd95476cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32fc65594f2151899374acf2c7cf1a40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://img.xkw.com/dksih/QBM/2012/3/20/1570811241472000/1570811247280128/STEM/3bb7730a3928426e89459eab1ee1f473.png?resizew=194)
(1)写出图中三对相似三角形,并对其中一对作出证明;
(2)连结
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31eee31191f484a7c5b391cde9f19ed2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
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10-11高二下·辽宁抚顺·期末
8 . 如图,△ABC内接于⊙O,过点A的直线交⊙O于点P,交BC的延长线于点D,且
AB2=AP•AD.
(1)求证:AB=AC;
(2)如果∠ABC=60°,⊙O的半径为1,且P为
的中点,求AD的长.
AB2=AP•AD.
(1)求证:AB=AC;
(2)如果∠ABC=60°,⊙O的半径为1,且P为
![](https://img.xkw.com/dksih/QBM/2011/7/14/1570257915494400/1570257920811008/STEM/45d76139383a47e49c29b4c031b66e6c.png?resizew=18)
![](https://img.xkw.com/dksih/QBM/2011/7/14/1570257915494400/1570257920811008/STEM/f712f2fd5143465aae89e8550e2ad88b.png?resizew=198)
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11-12高三上·山东聊城·阶段练习
9 . 如图,
是内接于
,
,直线
切
于点
,弦
,
与
相交于点
.
(I) 求证:
≌
;
(Ⅱ)若
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3947b65d110d74021faf8156b2f4d709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(I) 求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc3814287dbb60d478bffc5366f9928b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9f8f01137e92c0f2e63467036ae9cce.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19592a5eaacb7752f792d43652b43db8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://img.xkw.com/dksih/QBM/2011/12/25/1570632677564416/1570632682971136/STEM/8d453e2dd3474110b2e9f240122c8547.png?resizew=141)
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