tag:blogger.com,1999:blog-6933544261975483399.post3064718950467032999..comments2022-11-29T01:21:52.908-08:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1243: Quadrilateral, Four Exterior Angle Bisectors, Semi-sum, Angles. Mobile AppsAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-6933544261975483399.post-54782518169511198942016-08-03T12:51:21.807-07:002016-08-03T12:51:21.807-07:00https://goo.gl/photos/ZXhcHx3B542E7nND7
Let BC mee...https://goo.gl/photos/ZXhcHx3B542E7nND7<br />Let BC meet AD at P and AB meet CD at Q<br />In triangle CDP, CC1 and DC1 are angle bisectors of angles C and D<br />So PC1 is an angle bisector of angle P<br />In triangle ABP, BA1 and AA1 are external angle bisectors of angles B and A<br />So PA1 is an angle bisector of angle P => P, C1 and A1 are collinear<br />Similarly Q, B and D are collinear.<br />In triangles OCP and OCQ we have β = angle (PCQ)= θ+x+y<br />In triangles OAP and OAQ we have α= angle ( BAD)= θ-x-y<br />Add above expressions side by side we have θ= ½( α+ β)<br />Peter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-34884095485530491932016-08-02T23:49:38.834-07:002016-08-02T23:49:38.834-07:00Let BC and AD when extended meet at P. We have sho...<br />Let BC and AD when extended meet at P. We have shown earlier (see my proof<br />of Problem 1242) that A1,C1,P are collinear points and that this line is the bisector of <P = p say.<br /><br />Consider Tr. PCC1 where <OC1B1 = p/2 + (90 - â/2)..…(1)<br />Consider Tr. PAA1 where <OA1D1= (90 – á/2) – p/2….(2)<br /><br />A1B1C1D1 being cyclic (Problem 1241), from (2), <OB1C1 = (90 – á/2) –<br />p/2….(3)<br /><br />Now consider Tr. OB1C1 where è + {(90 – á/2) – p/2} + { p/2 + (90 - â/2)} =<br />180 from (1) and (3)<br /><br />Simplifying, è = (á+â)/2<br /><br />(Sorry about the different symbols for alpha, beta and theta)<br /><br />Sumith Peiris<br />Moratuwa<br />Sri LankaSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.com