tag:blogger.com,1999:blog-6933544261975483399.post3641253049145641398..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1322: Triangle, Angle Bisector, Circumcircle, Chord, Secant, Sum of two AnglesAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-6933544261975483399.post-4892512155890428742017-04-05T02:18:46.578-07:002017-04-05T02:18:46.578-07:00Easy to see, ABXZ, ABYW are cyclic, hence <AZX=...Easy to see, ABXZ, ABYW are cyclic, hence <AZX=<AWY=180-<ABC ( 1 ). Since <AZX=<AWX+<ZXW+<ZAW, we get the required <XWY=<ZXW+<ZAW.Stan FULGERhttps://www.facebook.com/stan.fulgernoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-79623064816928228312017-04-03T20:47:56.941-07:002017-04-03T20:47:56.941-07:00Easy to observe A,B,Y and W are concyclic.
Denote ...Easy to observe A,B,Y and W are concyclic.<br />Denote <ABW = <AXZ = <AYW = γ<br />Let AY and XW intersect at V.<br />We evaluate <VYW in two ways:<br />As an exterior angle of triangle VYW (i.e.) θ+γ<br />As an exterior angle of triangle VXA (i.e.) (α+β)+γ<br />Follows θ=α+βPravinhttps://www.blogger.com/profile/05947303919973968861noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-20593099444717958402017-03-13T21:59:43.251-07:002017-03-13T21:59:43.251-07:00Draw WV // ZX (with V on BC).
So <XWV = β (alte...Draw WV // ZX (with V on BC).<br />So <XWV = β (alternate angles).<br />Enough to show <YWV = α.<br />A,B,Y,W are concyclic (Note <WAY = <ZAX = <ZBX = α).<br />So <YVW = <YXZ = <BAY = <BWY.<br />Hence Triangles YVW, YWB are /// and <YWV = <WBY = α.<br />Pravinhttps://www.blogger.com/profile/05947303919973968861noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-80466731893998520092017-03-12T22:32:35.403-07:002017-03-12T22:32:35.403-07:00Problem 1322
let's say that <BAX=δ=<X...Problem 1322<br />let's say that <BAX=δ=<XZB(B,X,Z,A=concyclic) and <ΒWX=γ,but <XBZ=α=<YBW=<XAZ=<ZAW=<YAW.So the point A,B,Y and W are concyclic.Τhen<br /><BWY=<BAY or γ+θ=δ+α (δ=β+γ in triangle ZXW) or θ=α+β.<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-68342295941051792352017-03-12T16:21:49.029-07:002017-03-12T16:21:49.029-07:00Draw circle O1 throw A,W,Y,B. Extend AX to P, WX t...Draw circle O1 throw A,W,Y,B. Extend AX to P, WX to Q (P,Q on O1)<br />Ang PWY = α, need to prove PWQ = β<br /> ZX//WP => <PWQ = βc.t.e.onoreply@blogger.com