最新初中数学思维技巧专项训练(十三) 圆与相似结合综合题

圆与相似结合综合题类型一圆的基本质与相似的结合Y1⊙OABO∠∠ACB⊙OEABDS

∶ADE

()YA1B1∶3C∶2∶13YEABCAED.∽ABD(2)DEDB.Y13Y3AC⊙OBO∠30°.∠BDAC⊙OCD()(1)Y13Y4OBA⊙OBBDPCD∠PBC∠CBD(2)

BDY13Y5ABC⊙OACDEOB⊙OFBCBOFOFBGAB4Y类型二圆的切线相似的结合13Y6AB⊙OBCAB⊙OC⊙OABDADEE∥DCFAF.FG(2)

Y13Y7⊙OODF⊙OCE∠AEC∠ODCCD⊙OAB5CDY13Y8ABACOBCDE.DDFABFDEDF⊙OAE7ACYYAD∠BCDDAFFBFC.∠∠FCB12ABFA2YADBDADBDY⊙OCCDA∠(1)CD⊙O(2)B⊙OCD6BE13Y11Y∠ABA⊙OBCDABEFG(1)∠CAB(2)OHADHFHDGDFDH⊙O13YADEAErADEAEr详解D[]OE⊙Or∵∠AED∠∠ADE∠∴△ADE∵AB⊙O∴∠ACB∵∠B30°∴3r.∵CE∠∴∠ACE45°∴∠AOE∴AE2r.∴∶

△

(

()2∶CBrD.(1)E∴∠BAD∠∵∠∠∴∠BAD∠∵∠∠ADB∴△∽△.(2)∵∴∠∠.∵∠∠∴∠BAD∠∠∠CBD∵∠BAD∠∠∠∠CBD∠∴∠DBE∠∴DB(2)OD⊙OrABECDEDEC∴△∽△.∵AC⊙O∴∠ABCABC∵∠ACB30°()2(2△ABE()2(2△ABE∴ABACr.∵∠∴∠ABD45°∴∠AOD90°.ODC

ODOCr∴

SABr1)SCD2r(1)OC∵PC⊙O∴OC⊥∠OCP∵⊥PD∴∠BDP90°∴∠∠∴OC∥∴∠∠CBD∵OB∴∠PBC∠BCO∴∠PBC∠.(2)∵AB∴∠ACB90°∴∠ACB∠90°.∵∠ABC∠∴△ABC∽△∴

BCBDBCABBD.eq\o\ac(△,Rt)OFA∴AF∴∠AOF∠∵∠ABC∠ABG90°∴∠AOF∠ABG∴∥∵AO∴∴BG∠(2)CBE∠OEF∠CEB∴△FOECBE(ASA)∴FO2∴

AB

2BC5.DB∵⊙O∴∠ADB90°∴∠90°∴∠∠ABC∵∠∠ACB∴△∽ABC∴

BCACBCDC52DC

(1)EF∥BCABBG∴EF⊥AD∵AD∴FAFD∴∠FAD∠D∵GB⊥AB∴∠GAB∠G∠D90°∴∠∠G∵∠∠∴∠∠G∴FG.(2)AC∵ABBG∴AC⊙O∵⊙OC∴∠90°∴∠ACB∠90°.∵∠ACB∠90°∴∠∠CAB∵∠∠G∴∠CAB∠G∵∠CBA∠ABG90°∴△ABC∽∴

∴ABBCBGOC∵∠∠ABC∴∠ABC∠.∵∠AEC∴∠ABC∠ODC∵OCOD⊥∴∠∠OBCOF3OF3∠OCB∠90°∴∠∠COD∴∠180°∠CODOCCD∵⊙O∴⊙O(2)⊙OOD⊥∴BFBC∵∴

OB

BF

(1)∠∵∠OFB∠∴△OFB∴

OFCFCD×2OBCF10∴CDOD∵AB∴∠B∠∵OD∴∠∠∴∠∠∴OD.∵DF⊥∴ODDF.∵⊙O∴DF⊙O(2)∵⊙O∴∠AED∠180°.∵∠AED∠∴∠BED∠.∵∠B∠∴△BED△BCA∴

BDABBC∵OD∴BC3.∵∴

BE2(∴BE72∵AFBC∴∠FBC∠FAC180°.∵∠∠FACFA3FA3∴∠FBC∠CAD.∵∠EAC∴∠EAD∠∵∠EAD∠FAB∴∠∠∵∠FABFCB∴∠FBC∠FCB.(2)(1)∠FBC.∵∠FCB∠FAB∴∠∠∵∠∠∴△AFB∽△BFD∴

FBA∠FD∴BFFD12∴BF2∵FA2∴FD∴4.∵AB∴∠∠90°∴tan∠3∴∠FBAADAD∵∠∠30°∴CDAD·cos30°

23.OD∵OBOD∴∠OBDODB∵∠∠∴∠∠ODB∵⊙O∴∠ADB90°()∴∠ADOODB90°∴∠ADOCDA90°∠90°∴ODCD∵⊙O∴⊙O((2)∵∠∠C∠∠CBD∴△∽∴∵

ADBCBDBCBD∴CD∵BE⊙O∴BEDEBE⊥BC∴BE2BE(4BE)2

11OD∵BC⊙O∴⊥∵∠C90°∴ODAC∴∠∠ODA∵OAOD∴∠OAD∠∴∠∠OADADCAB(2)①DFDH.∵∠∴∠∠EFH.∵∠∠EAD∠HAF∴∠∠EFH