2025年中考二轮专题复习-几何压轴题高效拆分特训
专题四 辅助圆
INCLUDEPICTURE"标1.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标1.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标1.tif" \* MERGEFORMATINET 特训22 定点定长型INCLUDEPICTURE"标2.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标2.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标2.tif" \* MERGEFORMATINET
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 模型解读
图示 INCLUDEPICTURE"SK18-17.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-17.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-17.tif" \* MERGEFORMATINET INCLUDEPICTURE"SK18-18.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-18.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-18.tif" \* MERGEFORMATINET
条件 OA＝OB＝OC＝OD. O为定点，A为平面内一动点，且满足OA＝a.
结论 A，B，C，D四点在以点O为圆心的圆上. 点A在以点O为圆心，a为半径的圆上.
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 典题训练
1．如图，已知AB＝AC＝BC＝AD，则∠BDC＝________．
2．如图，在四边形 ABCD 中，DC∥AB，BC＝1，AB＝AC＝AD＝2，则 BD的长为________．
3．如图，OA⊥OB，P，Q分别是射线OA，OB上两个动点，C是线段PQ的中点，且PQ＝4，则在线段PQ滑动的过程中，点C运动形成的路径长为________．
4．如图，在矩形ABCD中，AB＝4，AD＝8，E是AB的中点，F是BC上一点，把△BEF沿着EF翻折，点B落在点B′处，则DB′的最小值为________．
5．如图，四边形ABCD是矩形．求证：A，B，C，D四点在同一个圆上．
INCLUDEPICTURE"SK18-23.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-23.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-23.tif" \* MERGEFORMATINET
INCLUDEPICTURE"标1.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标1.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标1.tif" \* MERGEFORMATINET 特训23 定弦定角型INCLUDEPICTURE"标2.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标2.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标2.tif" \* MERGEFORMATINET
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 模型解读
条件 图示及结论
在△ABC中，AB＝a(定长)，∠ACB＝α(定角)，探究点C的运动路径. INCLUDEPICTURE"SK18-24.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-24.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-24.tif" \* MERGEFORMATINET ①当α<90°时，点C的运动路径是优弧，∠AOB＝2α. INCLUDEPICTURE"SK18-25.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-25.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-25.tif" \* MERGEFORMATINET ②当α＝90°时，点C的运动路径是以AB为直径的⊙O(点C不与A，B重合). INCLUDEPICTURE"SK18-26.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-26.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-26.tif" \* MERGEFORMATINET ③当α>90°时，点C的运动路径是劣弧，∠AOB＝360°－2α.
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 典题训练
(1)【初步尝试】如图，已知线段AB和直线l，用直尺和圆规在l上作出所有的点P，使得∠APB＝30°(不写作法，保留作图痕迹)；
INCLUDEPICTURE"SK18-27.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-27.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-27.tif" \* MERGEFORMATINET
(2)【拓展延伸】如图，已知矩形ABCD，AB＝2，BC＝m，M为CD上的点．若满足∠AMB＝45°的点M恰好有两个，则m的取值范围为________；
INCLUDEPICTURE"SK18-28.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-28.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-28.tif" \* MERGEFORMATINET
(3)【迁移运用】
①如图①，在△ABC中，∠BAC＝45°，AD是BC边上的高，且BD＝6，CD＝2，则AD的长为________；
INCLUDEPICTURE"SK18-29.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-29.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-29.tif" \* MERGEFORMATINET
②如图②，在Rt△ABC中，AB⊥BC，AB＝8，BC＝6，P是△ABC内部的一个动点，满足∠PAB＝∠PBC，连接CP，则线段CP长的最小值为________．
INCLUDEPICTURE"标1.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标1.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标1.tif" \* MERGEFORMATINET 特训24 四点共圆INCLUDEPICTURE"标2.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标2.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标2.tif" \* MERGEFORMATINET
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 模型解读
条件 在△ABC和△ABD中，∠ACB＝∠ADB＝90°. 在四边形ABCD中，∠A＋∠C＝180°. 点C，D在AB的同侧，∠C＝∠D.
图示
结论 点A，B，C，D在同一个圆上，AB为该圆的直径. 点A，B，C，D在同一个圆上.
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 典题训练
1．如图，在四边形ABCD中，∠ABC＝∠ADC＝90°，∠CAD＝16°，AD＝BD，则∠ADB的大小为________°.
INCLUDEPICTURE"SK18-34.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-34.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-34.tif" \* MERGEFORMATINET (第1题) INCLUDEPICTURE"SK18-35.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-35.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-35.tif" \* MERGEFORMATINET (第2题)
2．如图，∠ACB＝∠ADB＝90°，AD＝BD，若AC＝2，BC＝4，则CD＝________．
3．如图，已知正方形ABCD的边长为6，点P是AB边上的一个动点，连接CP，过点P作PC的垂线交AD于点E，以PE为边作正方形PEFG，顶点G在线段PC上，对角线EG，PF相交于点O.当点P从点A运动到点B时，点O也随之运动，求点O经过的路径长．
INCLUDEPICTURE"SK18-36.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-36.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-36.tif" \* MERGEFORMATINET
INCLUDEPICTURE"标1.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标1.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标1.tif" \* MERGEFORMATINET 特训25 点圆、线圆最值INCLUDEPICTURE"标2.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标2.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标2.tif" \* MERGEFORMATINET
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 模型解读
点圆 条件：如图，M是半径为r的⊙O上的一动点(OM＝r定值)，点P是定点(OP＝d定值)．结论：INCLUDEPICTURE"SK18-37.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-37.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-37.tif" \* MERGEFORMATINET PM的最小值＝PM1＝|d－r|，PM的最大值＝PM2＝d＋r.
线圆 INCLUDEPICTURE"SK18-38.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-38.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-38.tif" \* MERGEFORMATINET 条件：如图，OP＝a，Q是一动点，且OQ＝b(a，b是定长，且a>b)．结论：点Q在以b为半径的⊙O上，且当PQ与⊙O相切时，∠OPQ最大.
INCLUDEPICTURE"SK18-39.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-39.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-39.tif" \* MERGEFORMATINET 条件：如图，O是直线l上的一定点，P是l上的一动点，OQ＝m(定长)，且∠OPQ＝α(定角)．结论：点Q在以m为半径的⊙O上，且当PQ与⊙O相切时，OP最大.
INCLUDEPICTURE"条单.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\条单.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\条单.tif" \* MERGEFORMATINET 典题训练
1．如图，在平面直角坐标系xOy中，直线AB经过点A(6，0)，B(0，6)，⊙O的半径为2(O为坐标原点)，点P是直线AB上一动点，过点P作⊙O的一条切线PQ，Q为切点，则切线长PQ的最小值为( )
INCLUDEPICTURE"SK18-40.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-40.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-40.tif" \* MERGEFORMATINET
A. B．3 C．3 D.
2．[2024宁德质检4分]如图，正方形纸片ABCD的边长为4，点E在AD边上，点F在CD边上，将正方形纸片ABCD沿EF对折，点B的对应点是点G，连接DG，若AE＝1，则DG长的最小值是________．
INCLUDEPICTURE"SK18-41.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-41.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-41.tif" \* MERGEFORMATINET
3．如图，在矩形ABCD中，AB＝4，若在边CD上存在一点P，使得∠APB＝90°，则矩形ABCD面积的最大值为________．
INCLUDEPICTURE"SK18-42.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\SK18-42.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SK18-42.tif" \* MERGEFORMATINET
专题四 辅助圆 答案
INCLUDEPICTURE"标1.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标1.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标1.tif" \* MERGEFORMATINET 特训22 定点定长型INCLUDEPICTURE"标2.tif" INCLUDEPICTURE "D:\\课件\\中考福建数学\\标2.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\标2.tif" \* MERGEFORMATINET
1．150° 2. 3.π 4.2－2
5．证明：如图，连接AC，BD，交于点O，
INCLUDEPICTURE"SKD-18.tif" INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SKD-18.tif" \* MERGEFORMATINET
∵四边形ABCD是矩形，
∴OA＝OB＝OC＝OD，
∴A，B，C，D四点在以点O为圆心，
AC长为半径的圆上．
即A，B，C，D四点在同一个圆上．
特训23 定弦定角型
解：(1)如图所示，点P1、P2即为所求．
INCLUDEPICTURE"SKD-19.tif" INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SKD-19.tif" \* MERGEFORMATINET
(2)2≤m<＋1
(3)①4＋2 ②2－4
特训24 四点共圆
1．32 2.2
3．解：如图，连接OA，OC，
INCLUDEPICTURE"SKD-20.tif" INCLUDEPICTURE "C:\\Users\\庞建宇\\Desktop\\中考数学福建\\SKD-20.tif" \* MERGEFORMATINET
∵四边形ABCD是正方形，
∴∠B＝90°，∠BAC＝45°，
∴AC＝＝＝6，
由题易知A，P，O，E四点共圆，
∴∠OAP＝∠OEP＝45°，
∴点O在AC上．
当点P运动到点B时，O为AC的中点，
∴OA＝AC＝×6＝3，
∴点O经过的路径长为3.
特训25 点圆、线圆最值
1．D 2.－3 3.8
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