第二十四章圓知識點及典型例題

第二十四章圓知識點及典型例題Chapter24:SummaryofConceptsandTypesofQuestionsonCirclesExperimentMiddleSchool,MaGuirongI.ConceptofCirclesInsetform:1.Acirclecanbeseenasasetofpointsthatareequidistantfromafixedpoint.2.Theoutsideofacirclecanbeseenasasetofpointsthatarefartherthanafixedlengthfromafixedpoint.3.Theinsideofacirclecanbeseenasasetofpointsthatarecloserthanafixedlengthfromafixedpoint.Intrajectoryform:1.Acircleisthetrajectoryofpointsthatareequidistantfromafixedpoint.Knowinganytwoofthefollowingcanleadtothedeductionoftheotherthree:①ABisadiameter②ABisperpendiculartoCD③CE=DE④ArcBC=arcBD⑤ArcAC=arcAD2.Thetwoparallelchordsofacirclecutoffcongruentarcs.IncircleO,sinceAB||CD,arcAC=arcBD.Additionalconcepts:2.Perpendicularbisector:thetrajectoryofpointsequidistantfromthetwoendpointsofalinesegment.3.Anglebisector:thetrajectoryofpointsequidistantfromthetwosidesofanangle.4.Thetrajectoryofpointsequidistantfromalineistwoparallellinesthatareequidistantfromthelineandafixedlengthapart.5.Thetrajectoryofpointsequidistantfromtwoparallellinesisalinethatisparalleltothetwolinesandequidistantfromthem.II.PositioningofPointsandCircles1.Apointisinsideacircleifitsdistancefromthecenterislessthantheradius.2.Apointisonacircleifitsdistancefromthecenterisequaltotheradius.3.Apointisoutsideacircleifitsdistancefromthecenterisgreaterthantheradius.III.PositioningofLinesandCircles1.Alineisexteriortoacircleifitsdistancefromthecenterisgreaterthantheradius.2.Alineistangenttoacircleifitsdistancefromthecenterisequaltotheradiusandithasoneintersectionpoint.3.Alineintersectsacircleifitsdistancefromthecenterislessthantheradiusandithastwointersectionpoints.IV.PositioningofCircles(optional)1.Disjoint:nointersectionpoints,distancebetweencentersisgreaterthanthesumoftheradii.2.Externallytangent:oneintersectionpoint,distancebetweencentersisequaltothesumoftheradii.3.Intersecting:twointersectionpoints,distancebetweencentersisbetweenthedifferenceandsumoftheradii.4.Internallytangent:oneintersectionpoint,distancebetweencentersisequaltothedifferenceoftheradii.5.Contained:nointersectionpoints,distancebetweencentersislessthanthedifferenceoftheradii.V.PerpendicularityTheoremPerpendicularityTheorem:adiameterperpendiculartoachordbisectsthechordandthearcitsubtends.Corollaries:(1)Adiameterperpendiculartoachord(notadiameter)bisectsthechordandthetwoarcsitsubtends.(2)Theperpendicularbisectorofachordpassesthroughthecenterandbisectsthetwoarcsitsubtends.(3)Adiameterthatbisectsoneofthearcssubtendedbyachordisperpendiculartothechordandbisectstheotherarc.VI.CentralAngleTheoremCentralAngleTheorem:incongruentorsimilarcircles,congruentcentralanglescorrespondtocongruentchords,arcs,andchord-to-centerdistances.Knowinganyoneofthefourcorollariescanleadtothedeductionoftheotherthree.1.圓周角定理圓周角定理指出，同弧所對的圓周角等于它所對的圓心角的一半。例如，在圓O中，弧AB所對的圓心角為∠ACB，弧AB所對的圓周角為∠AOB，則有∠AOB=2∠ACB。此外，圓周角定理還有兩個推論。第一個推論是，同弧或等弧所對的圓周角相等；同圓或等圓中，相等的圓周角所對的弧是等弧。第二個推論是，半圓或直徑所對的圓周角是直角；圓周角是直角所對的弧是半圓，所對的弦是直徑。這些推論在解題時經常用到。2.圓內接四邊形圓的內接四邊形定理指出，圓內接四邊形的對角互補，外角等于它的內對角。例如，在圓O中，四邊形ABCD是內接四邊形，則∠C+∠BAD=180°，∠B+∠D=180°，且∠DAE=∠C。3.切線的性質與判定定理切線的判定定理指出，過半徑外端且垂直于半徑的直線是切線。例如，在圓O中，若MN過半徑OA外端且垂直于OA，則MN是圓O的切線。切線還有一個性質定理，即切線垂直于過切點的半徑。由此可推出兩個推論：一是過圓心垂直于切線的直線必過切點；二是過切點垂直于切線的直線必過圓心。這些定理和推論也稱為二推一定理，即知道其中兩個條件就能推出最后一個。4.切線長定理切線長定理指出，從圓外一點引圓的兩條切線，它們的切線長相等，這點和圓心的連線平分兩條切線的夾角。例如，在圓O外一點P處引切線PA、PB，則有PA=PB，且OP平分∠BPA。5.圓冪定理圓冪定理是一組關于圓和圓外點之間的長度關系的定理。其中，相交弦定理指出，圓內兩弦相交，交點分得的兩條線段的乘積相等。例如，在圓O中，若弦AB和CD相交于點P，則有PA×PB=PC×PD。5、如圖，F(xiàn)是以O為圓心，BC為直徑的半圓上任意一點，A是BC的中點，AD⊥BC于D，求證：AD=1/2BF.例題3、度數(shù)問題已知：在⊙O中，弦AB=12cm，O點到AB的距離等于AB的一半，求：∠AOB的度數(shù)和圓的半徑.例題4、平行問題在直徑為50cm的⊙O中，弦AB=40cm，弦CD=48cm，且AB∥CD，求：AB與CD之間的距離.例題5、同心圓問題如圖，在兩個同心圓中，大圓的弦AB，交小圓于C、D兩點，設大圓和小圓的半徑分別為a,b，求證：AD×BD=a^2-b^2.例題6、利用切線性質計算線段的長度如圖，已知：AB是⊙O的直徑，P為延長線上的一點，PC切⊙O于C，CD⊥AB于D，又PC=4，⊙O的半徑為3，求：OD的長度.例題7、利用切線性質計算角的度數(shù)如圖，已知：AB是⊙O的直徑，CD切⊙O于C，AE⊥CD于E，BC的延長線與AE的延長線交于F，且AF=BF，求：∠A的度數(shù).例題8、利用切線性質證明角相等如圖，已知：AB為⊙O的直徑，過A作弦AC、AD，并延長與過B的切線交于M、N，求證：