tag:blogger.com,1999:blog-6933544261975483399.post8129929045250300918..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Problem 871: Brahmagupta's Theorem, Cyclic Quadrilateral, Perpendicular Diagonals, MidpointAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6933544261975483399.post-41181820512054544942016-09-01T12:11:40.281-07:002016-09-01T12:11:40.281-07:00Problem 871
Let x=<BCA=<BDA=<AEF (perpe...Problem 871<br />Let x=<BCA=<BDA=<AEF (perpendicular line) or <MEC=x.Then ME=MC.<br />But <EBC=90-x=<BEM.So BM=EM.Therefore BM=MC.<br />APOSTOLIS MANOLOUDIS 4 HIGH SCHOOL KORYDALLOS PIRAEUS GREECE<br />APOSTOLIS MANOLOUDIShttps://www.blogger.com/profile/15561495997090211148noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-13017725279131593832013-04-25T07:48:19.962-07:002013-04-25T07:48:19.962-07:00Far more general:
If ABCD is cyclic quadrilateral...Far more general:<br /><br />If ABCD is cyclic quadrilateral and its diagonals cuts in E, then orthocenter of tr ABE, circumcenter of tr DEC and E itself are colinear.<br /><br />In this particular case, EF contains the orthocenter of tr AED so must contain the circumcenter of tr BEC, but this is M.<br />Done.Editorhttps://www.blogger.com/profile/18079120609888942700noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-16222088255109194262013-04-22T21:19:33.288-07:002013-04-22T21:19:33.288-07:00My solution http://www.docstoc.com/docs/153902983/...My solution http://www.docstoc.com/docs/153902983/Problem-871Bogdan Rusuhttps://www.blogger.com/profile/01815872777807580975noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-65761190424510135502013-04-20T02:15:09.287-07:002013-04-20T02:15:09.287-07:00Angle MCE = Angle EDF = Angle MED - 90 = Angle MEC...Angle MCE = Angle EDF = Angle MED - 90 = Angle MEC <br />So triangle MCE is isosceles and so as triangle BEM for symmetry. <br />Hence BM = EM = CM <br />Q.E.D.nakahttps://www.blogger.com/profile/11277356476170372732noreply@blogger.com