英语翻译1．Twoplanes,and,intersectalongtheline.Thepointisgivenintheplane,andthepointintheplane;neitherofthesepointsliesonthestraightline.Constructanisoscelestrapezoid(with)inwhichacirclecanbeinscribed,an

　　英语翻译

　　1．Twoplanes,and,intersectalongtheline.Thepointisgivenintheplane,andthepointintheplane;neitherofthesepointsliesonthestraightline.Constructanisoscelestrapezoid(with)inwhichacirclecanbeinscribed,andwithverticesandlyinginplanesandrespectively

　　2．Constructtriangle,given,(thealtitudesfromand),and,themedianfromvertex.

　　3．Constructatriangleif,and,whereisthemidpointofthesegmentand.Provethatasolutionexistsifandonlyif

　　4．Consideranisoscelestriangle.letbetheradiusofitscircumscribedcircleandbetheradiusofitsinscribedcircle.Provethatthedistancebetweenthecentersofthesetwocircleis

　　5．Onthecircletherearegiventhreedistinctpoints.Construct(usingonlyastraightedgeandacompass)afourthpointonsuchthatacirclecanbeinscribedinthequadrilateralthusobtained.

　　6．Pointandsegmentaregiven.Determinethelocusofpointsinspacewhichareverticesofrightangleswithonesidepassingthrough,andtheothersideintersectingsegment.

　　7．Acircleisinscribedinatrianglewithsides.Tangentstothecircleparalleltothesidesofthetrianglearecontructe.Eachofthesetangentscutsoffatriagnlefrom.Ineachofthesetriangles,acircleisinscribed.Findthesumoftheareasofallfourinscribedcircles(intermsof).

　　8．Considerwithacuteangle.Thorughapointperpendicularsaredrawntoand,thefeetofwhichareandrespectively.Thepointofintersectionofthealtitudesofis.Whatisthelocusofifispermittedtorangeover

　　a)theside;

　　b)theinteriorof.

　　9．Letbeatriangle,andlet,,bethreepointsintheinteriorsofthesides,,ofthistriangle.Provethattheareaofatleastoneofthethreetriangles,,islessthanorequaltoonequarteroftheareaoftriangle.

　　Alternativeformulation:Letbeatriangle,andlet,,bethreepointsonthesegments,,,respectively.Provethat

　　,

　　wheretheabbreviationdenotesthe(non-directed)areaofanarbitrarytriangle.

　　10．Theparallelogramhas,andthetrianglehasallanglesacute.Provethatcirclesradiusandcentercovertheparallelogramifandonly