1 . 已知如图,
是锐角
的角平分线,
,
,且
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb2cf0e95fdf1fd8a5b01d3dfd905e08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102e0f5ca723747618a309c8b303fff4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b75efb848c29d6e8ee81d1ee27edfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab6bc5042211aa0130d17fecc23936fe.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/6/9ab99742-136a-4064-9cce-1a9e6a2dd60e.png?resizew=194)
您最近一年使用:0次
2 . 已知距离为
的
、
两点在直线
的同侧,且
、
到直线
的距离分别为
、
.问能否作出经过
、
两点且与直线
相切的圆?若能,请写出作法,画图并求出圆的半径;若不能,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc31bf4b6ed8cf336432a5a2791e67e3.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/0f85fca60a11e1af2bf50138d0e3fe62.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/095ab4a92bf822e175d370e6d0c8a730.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d19daa2b38e3a52bd7096d235c843110.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/0f85fca60a11e1af2bf50138d0e3fe62.png)
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3 . 在锐角
中,
为外接圆半径.(1)求证:
;(2)当去掉锐角三角形的条件时,讨论命题的真假.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7acaf98c87c3d69d9a3162097ff61ebd.png)
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4 . 已知等腰
的顶角
为
,
是腰
上一点,且
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef6f7d09a282d4f8f16697d349ffaab6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db827cbb22a564c4dbaa6e1c78c6dae5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d2530e7023b2345c651e8f53629ff1.png)
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5 . 如图,半径为
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3e95410f3b4fcb0cba425b521d1f67.png)
的两圆交于
、
两点,
是半径为
的圆上任意一点(不在另一圆内),
、
与另一圆分别交于点
、
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/6/1958d588-2612-4639-8e57-d72f96541bfd.png?resizew=176)
(1)用
、
及
表示
的长度;
(2)证明:两圆正交(即交点处切线互相垂直)的充分必要条件为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3e95410f3b4fcb0cba425b521d1f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db70a10c871657f526e5a9acc5483b65.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/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd58415d4a40dfd7ae911997aba705a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44172d85b47b3626f565bad190d5553c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/6/1958d588-2612-4639-8e57-d72f96541bfd.png?resizew=176)
(1)用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3e95410f3b4fcb0cba425b521d1f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72689e44ff4e828835ee13ea82320fc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
(2)证明:两圆正交(即交点处切线互相垂直)的充分必要条件为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29750466989200e0d94ef2ad6caa0a3a.png)
您最近一年使用:0次
6 . 已知锐角
.以
、
、
为三边作一
,以
、
、
为圆心,分别以
、
、
为半径作圆,则此三圆必交于一点
,且
恰是
的垂心.试证之.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b01adc561735ff5be9bb97266918f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8e5d4f93699f8dcffb0e7840ca5597e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c32e2f2d7147cf1699fbfdef9cf4af74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a71399ec88d4a3bd807e23bef04dce6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5201fc26d013f6fb889933c0e32f5c53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1311f32edf13f8caee5edb03f24a7ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6402f1010e94be78552ed4c45548b1b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a71399ec88d4a3bd807e23bef04dce6.png)
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7 . 设凸六边形
的六个内角均为钝角,且
,
,
.以三边
为边向内各作一个矩形,使它们的顶点两两重合,构成
(如图)求证:矩形
的面积之比等于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb9878191cc9f39fb3441ed814ce56d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d87ab70c78afcfaea15b5e427a93cbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/816f8c66d0fbb89a0176eab213c6823d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c30ac5adb0b9c4419d61b68b72dbd8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14effb5263b6b8ed7748f05586177a63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cb46aadc161a00589001e120a2c5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2ad0a7e83e7e96f900bcb539f5afdac.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/7/ef72d742-233d-45f6-bd2f-9c3028544c3a.png?resizew=175)
您最近一年使用:0次
8 . 设
为
斜边
上的高,
、
、
分别是
、
、
的内心.求证:
的外接圆半径与
的内切圆半径相等.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87fd28b8b2bde9de5630a6106a6f762e.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/3c4f6f74444b2b7947fc6e35c8d62322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c4f6f74444b2b7947fc6e35c8d62322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9f8f01137e92c0f2e63467036ae9cce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df6d51738ac1bc8b9530ea4a55745c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31d8befc53da1c5fd89e032b8db550d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
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9 . 设K为△ABC的内心,点
、
分别为边AB、AC的中点,直线AC与
交于点
,直线AB与
交于点
,若
与△ABC面积相等,试求∠CAB.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f601f590f8a8247d126d77ae289f80c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd864e2c2b71121a8cadb61e7c7f0511.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba39ff97bcbed1e5ec9390d599b655b5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/6/c5d8bfb3-0c26-41b4-867d-08c917e12709.png?resizew=205)
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10 . 如图,在
中,
分别是
延长线上的点,
为
的中点,连结
交
外接圆于
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80d867269ef4acecaa6383ff237d3ae1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0d54431bbb28ebd98db5c1dc6083a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6795cae2df43a722e1355e9562d93c09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c09eec4e14a861af83d7828797d176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50ebdc58f86932d71ba699ac3cc57606.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/5/3d77283c-2898-4f5e-b943-7d1ae9bba616.png?resizew=137)
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