tag:blogger.com,1999:blog-6933544261975483399.post6412586084145097436..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1258: Cyclic Quadrilateral, Concyclic Points, MeasurementAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6933544261975483399.post-12531140838996991932016-09-11T22:59:48.120-07:002016-09-11T22:59:48.120-07:00Problem 1258
Let the point K on the side AE (no...Problem 1258<br /> Let the point K on the side AE (no extension) such that AK=BC=4.So KE=1,triangle AFK=triangle BDK(AF=BD,AK=BC,<FAK=<DBC).So CD=KF then <AKF=<BCD=180-<FAG=180-<FKE. So <FAG=<FKE .But <FGA=<GFA=<FEA=<FEK,so <KFE=<KEF(<KFE=180-<FKE-<FEK=<br />180-<FAG-<FGA=<AFG).So KF=KE=CD=1.<br />APOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORYDALLOS PIRAEUS GREECE<br />APOSTOLIS MANOLOUDIShttps://www.blogger.com/profile/15561495997090211148noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-31392962014622354132016-09-10T21:16:10.817-07:002016-09-10T21:16:10.817-07:00https://goo.gl/photos/ffCv5yAwZMzaTRv7A
Connect F...https://goo.gl/photos/ffCv5yAwZMzaTRv7A<br /><br />Connect FG, EF<br />Extend BC to H such that CH=CD=x ( see sketch)<br />We have AG=AF => ∠ (AGF)= ∠ (AFG)= u<br />Let ∠ (GAE)= w => ∠ (BDC)= ∠ (BAC)= ∠ (GFE)= w<br />Let ∠ (CAD)= ∠ (DBC)=v<br />Since ∠ (DCH)= ∠ (BAD) => Isoceles triangles GAF simillar to triangle HCD ( case AAA)=> ∠ (CDH)= u<br />Triangle AFE congruent to BDH ( case ASA)<br />So AE= BH=> 5=4+x => x=1<br />Peter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-11347284524686026602016-09-10T17:26:27.074-07:002016-09-10T17:26:27.074-07:00Let GE = p, EF = q and FG = r
Let AG = AF = BD = y...Let GE = p, EF = q and FG = r<br />Let AG = AF = BD = y<br /><br />Tr.s GEF and BCD are similar so, <br /><br />p/4 = q/x = r/y .........(1)<br /><br />Applying Ptolemy to cyclic quad AGEF, <br />py + qy = 5r and hence from (1). <br /><br />4r + xr = 5r which yields <br />x = 1<br /><br />Sumith Peiris<br />Moratuwa<br />Sri LankaSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.com