tag:blogger.com,1999:blog-6933544261975483399.post3579618365407063347..comments2023-02-02T05:46:39.693-08:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1300: Arbelos, Semicircles, Diameters, Circle, Incircle, Tangent, Perpendicular, Concurrent LinesAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-6933544261975483399.post-75918467452634771852016-12-31T11:46:59.392-08:002016-12-31T11:46:59.392-08:00https://goo.gl/photos/t4BLKDL5KmEciriP8
Draw poin...https://goo.gl/photos/t4BLKDL5KmEciriP8<br /><br />Draw points D, E , F as per sketch<br />Per the result of problem 638 quadrilaterals ATT2B and CTT1B are cyclic with centers at D and E.<br />Per the result of problem 1298 T, P and B are collinear<br />Perform geometry inversion center C , power of inversion= CB.CA circle E will become line AF<br />T1 is the intersection of circles O1 and E so the image of T1 is the intersection of circle O1 and line AF which is point D.<br />So C, T1 and D are collinear<br />Similarly we also have A, T2, E are collinear.<br />Let DT1 cut circle I at P’ <br />Triangles O1DT1 and IT1P’ are isosceles and similar ( case AA)<br />So ∠DO1T1=∠T1IP’ => IP ‘⊥AC => P’ coincide to P <br />With similar way ET2 will cut circle I at P<br />So TB, IH, CT1 and AT2 are concurrent<br /><br />Peter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-80368841311735683062016-12-31T05:15:16.430-08:002016-12-31T05:15:16.430-08:00Problem 1300
According to the problems of 1168 an...Problem 1300<br />According to the problems of 1168 and 1298, K medium the arc AB and L medium the arc BC. Is K the center the circle passing through the points A,B,T_2 and T, and L is center ( T,T_1,B,C).<br />The TB intersects IH in P in the circle (Ι,ΙΤ).Ιf AP intersects the circle<br />(I,IT) and in T’, then <PTT’=<PIT’/2=90-<IPT’=90-<APB=<BAT’ so the points T,T’ ,B and A are concyclic.Then the points T’ and T_2 coincide.<br />So the points A,P and T_2 are collinear.Similar the points T_1,P and C are<br />collinear. <br />APOSTOLIS MANOLOUDIS KORYDALLOS PIRAEUS GREECE<br /><br />APOSTOLIS MANOLOUDIShttps://www.blogger.com/profile/15561495997090211148noreply@blogger.com