直线和圆的位置关系典型习题.doc

直线和圆的位置关系典型习题1.如图，AB为O的直径，C为O上一点，AD和过点C的切线垂直于D， BE和过点C的切线垂直于E求证：（1）AC平分DAO（2）BC平分EBO（3）DC = CE（4）AD + BE = AB（5）ADCACB（6）CEBACBABCDEO（7）AC2 = ADAB（8）BC2 = BEAB2.已知：PA、PB分别切O于A、B，C为优弧上一点，D为劣弧上一点。求证：（1）D = 90+ P（2）C = 90 PPABCDAB3.如图，两个同心圆，大圆的弦AB和小圆相切，AB = a求证：圆环的面积 = 4.已知：AB为O直径，ADBC，B = 90，DC切O于E求证：（1）CD = AD + BC（2）COD = 90（3）OE2 = ADBCOABCDEFG （4）连结AC、BD，交于点F，则EFBCABCDEO （5）延长EF交AB于G，则GF = EF5.RtABC中，B = 90，AB = 6，BC = 8，则ABC的内切圆半径为_，ABC的外接圆半径为_。ABPCDEO6.已知PA、PB分别切O于A、B，E为劣弧AB上一点，过E点的切线交PA于C、交PB于D。（1） 若PA = 6，求PCD的周长。（2） 若P = 50求DOC7.已知圆外切四边形ABCD，则ABBCCDDA = 645_。（已知圆内接四边形ABCD，则ABCD = 234_）8.已知ABC中，AB=AC=13，ABC的面积为60，求ABC内切圆直径（此题两个结果，可用相似求，也可利用S=Cr ）9.已知：如图，PA、PB分别切O于A、B，AC为O直径求证：（1）POBC（2）ABCPBO（3）POBC = 2OB2（ABPOCE4）（5）E为PAB的内心DABCEI10.如右图，点I是ABC的内心，AI交边BC于点D，交ABC的外接圆于点E。求证：（1）BE = IE（2）IE2 = DEAE11、等边三角形的边长为，则它的内切圆半径为_，外接圆半径为_，高线为_。12、在ABC中，A的平分线AE交BC于D，交ABC的外接圆于E。（1）连结BE、CE，图中有哪些相等的角？有哪些相等的线段？并证明：ADCBDEABEABDCEDAEC（2）过E作圆的切线MN，交AB的延长线于M，交AC的延长线于N，证明：MNBC（3）过B作圆的切线BF，证明：BE平分DBFDABCEFDABCEG（4）过A作圆的切线AG，交BC的延长线于G， 证明：AGD为等腰三角形。DABCEMNDABCE(1) (2) (3) (4)13、已知：AD切ABC的外接圆于A，交BC的延长线于D，DF平分ADB，分别交AC、AB于E、F。求证：（1）DCDB = AC2AB2（2）AE = AFDABCEF（3）图中共有多少组相似三角形（4）AE2 = CEBE14、过O内部或外部的一点P作O的割线，交O于A、B两点。求证：PAPB = OP2 r2（r为O的半径）PPOABABO15、半径为5的O中，半径OC垂直于直径AB，割线PC交O于D，交AB的延长线于P，若CD = 2，分别求PC、PB、BD的长（两种方法：列方程，作OEPC于E）OABCPD16、已知：AB为O直径，BAC = 20，D为的中点， ED切O于D，求EDAABCDE17、延长O的直径AB到P，使PB = OB，过P作PC切O于C，过B作BD切O于B，交PC于D。OABPCD（1） 求P、A的度数（2） 求CDDP18.已知AB是O的直径，BC是O的切线，切点为B，OCAD，（1）求证：DC是O的切线，（2）若O的半径为R，求证：ADOC=2 （3）若BC=6，O的半径为3.求AD的长。（4）设OC交O于N，当AE：AO等于多少时，四边形AOND菱形，请说明理由。 19.如图ABC中，AB=AC，以AB为直径的O交BC于D，DEAC于E，交BA的延长线于H。（1）求证：DE是O的切线，（2）若AH=，tanFBD=,求O的直径。（3）试探究四边形AEFG的形状。BD，BA， BF三条线段之间的关系式，过H作DHB的平分线交BC于M，当AB的长度变化时H在直线AB上的位置也发生变化，试问HMD的大小是否发生变化，若不发生变化，求出HMD的度数；若发生变化，请说明理由。（4）AH=，tanFBD= , 求BG的长度. 20.如图以RtABC的直角边AC为直径的O交斜边AB于D，过O作OEAB，交BC于E（1）求证：DE是O的切线，（2）CE=BE（3）如果O的半径为，ED=2求AB的长。（4）当BAC为多少度时，AOED为平行四边形？（5）在（4）的条件下，探索CODE的形状。21.如图AB=AC , ABC =,以AC为直径作O，CO与O交于D，AD交BC于E，F是BE的中点，（1）判断DF与O的位置关系。（2）下列四个结论=CECB 4=EDEA OCB=EAB DF=CD 其中正确结论是 （只填序号）并选择一个证明22.已知点A，B，C，D在O上，DA=DC，AC，BD相交于E，DE=EB，延长BA至F，使AF=AB，连接DF（1）BD平分ABC（2）若DE=3，求AB的长 3试判断DF与O的位置关系（4）求F的正弦值 （5）探索ABCD与BODC的形状23、 如图AB是O的直径，AC=CE，CDAB于D，AE交CD于F，交BC于K。（1）求证：AF=CF （2）若tanECA=CD=2，求O的直径 （3）连接OF，试判断OF与BK之间的关系 （4）写出图中至少三个不同的比例式或乘积式，并选择一个证明。24.在ABC中AB=AC，O为AB的中点，以O为圆心，OB为半径的圆交BC于D，过D作DEAC于E （1）求证：DE是O的切线 （2）若点O沿AB向点B运动，其它条件不变，那么DE与O有什么位置关系？并证明（3）若AB=AC=5，SinA=，那么圆心O在AB的什么位置时，O与AC相切？请说明理由8知ABC内接于以AB为直径的O过C作O的切线交AB的延长线于D，DA：AB=1：2 （1）求CDB的度数 （2）在切线DC上截取CE=CD，连接EB，判断EB与O有什么位置关系？并证明 （3）利用图中已标明的字母，连接线段，指出至少5对相似三角形（不包含全等，不需要证明）25如图A为O的直径EF上的一点，OB是和这条直径垂直的半径，BA和O相交于另一点C，过C的切线交EF的延长线于D，（1）求证：DA=DC AbxAC=AExAF (2)当DF：EF=1：8且DF=时，求ABxAC的值 （3）现将直径EF所在的直线向上平移到如图所示的位置其它条件不变，试猜想DA=DF是否仍然成立？26、如图1，AB,AC分别为o的直径和弦，D为的中点，过点D的切线与AC的延长线相交于点E。求证：DEAC； 求证：AD2=AEAB；试判断EA,EC,ED三条线段之间的数量关系，并证明；过点D作DFAB于F，如图2， 求证：ECEAAFBF； 27、如图1，已知AB是o的直径，延长AB到D，过D作o的切线DC，C为切点。求证：DC2=DADB；如图2，求证：；若DC6，求AB的长；如图3，过点C作CGAB于F，交o于G，若，BD=9,求 o的半径及SinD； 28、已知：AB是o的直径，BC是切线，切点为B，DC是o的切线。求证：OCAD； 若圆的半径为R, 求证：ADCO2R2；在的条件下，若OA2，ADOC6，求CD的长；延长 CD，BA相交于点E，若EA1，ED2，求tanEDA；若AD2,DC，求tanEDA；若AE3,CD:DE2：3，求 o的半径；若BC6, o的半径为3，求AD的长；若ABBC2, 求ED:EC；29、如图，在ABC中，以BC为直径的O交AB于点F，E为AC上一点，EF是O的切线，E为AC的中点。1求证：AC是O的切线；2求证：AFAB=4EF2 ；3连接OE，若四边形OEFB是平行四边形，求SinABE的值； 30、ABC是o的内接三角形，弦AD平分BAC，交BC于点E 。求证：ABACADAE；求证：BD2DEDA；求证：AD2BD2ABAC；求证：AE2BEECABAC； 31、如图，四边形ABCD内接于o，过点C的直线分别交AB、AD于F、H，AC平分BAD，HFBD。求证：FH是o的切线；求证：BC2CECA；求证：CD2ABDH；若I是ABD的内心，试判断CI、CE、CA之间的数量关系，并证明；32ABAC,以AB为直径O交BC于点D，交AC于点E。1求证：DEBC。2已知AB5cm，DE3 cm，求BE的长。3过点D作DFAC,求证：F是CE的中点。4在的条件下，求证：BC24CFCA。5若ABBC56,AE7,求CF的长。 33、已知ADBC,BCD900,以AB为直径的O与DC相切于点E，交BC于F。1求证：AE平分BAD。2求证：AD+BC=AB。3求证：点E是DC的中点。4试判断AD与FC的关系，并证明。5求证：DC2=4ADBC。6试判断AD，AE,AB三条线段之间数量的关系，并证明。34、如图，割线ABC与O相交于B,C两点，D为O上一点，ACOE于F，DE交AC于G，ADG=AGD,1求证：AD是O的切线。2试探究线段AD,AB,AC之间的数量关系，并证明。3若AB=2,AD=4,EG=2, 求O的半径。35、如图BC为半园O的直径，ADBC,=,BF和AD交于E。1求证：AB2=BEBF。2求证：=。3连接OE,试判断OE与MC的关系，并证明。 36、ABC是O的内接三角形，D是的中点，延长DA,CB相交于点G。1求证：ABACADAG。2若AC为O的直径，EAG1200,BC6,求AG的长。3在的条件下，求证：BC2DADG。37、已知，o的内接四边形是ABCD，对角线AC与BD互相垂直于点E。1连接AO并延长，交o于点F，求证：ABADAEAF。2若AF与BD相交于点M，求证：ABBFACBM。3过点O作ONAB于N，试判断ON与CD之间的数量关系，并证明。 38、如图，PA为O的切线，PBC为割线，APC的平分线交AB于E，交AC于F,交O于Q。(1) 求证；PAE=C。 （2）求证：AE=AF。(3)求证： = 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