tag:blogger.com,1999:blog-6933544261975483399.post3186181811773312317..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1110: Right Triangle, External Squares, Catheti, Cathetus, Angle Bisector, 45 Degree, Internal SquareAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6933544261975483399.post-75150756278301530182015-12-20T13:25:36.915-08:002015-12-20T13:25:36.915-08:00Although similar to Jacob Ja's, CQ/AC=BC/(BC+A...Although similar to Jacob Ja's, CQ/AC=BC/(BC+AB)=BC/CD=CN/CE, hence the triangles ADE and QBN are homothetic, therefore similar; likewise CFG and QMB, therefore we are done.<br /><br />Best regardsStan Fulgerhttps://www.facebook.com/stan.fulger?fref=ufinoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-2237134957183669612015-04-19T10:34:55.671-07:002015-04-19T10:34:55.671-07:00We call BN=Z ; BM=Y NQ=X, AB=c; BC=a
First step:...We call BN=Z ; BM=Y NQ=X, AB=c; BC=a<br />First step: We have: Z/c=a/(a+c) hence Z= ac/(a+c)<br />Y/a=c/(a+c) hence Y=ac/(a+c) so Y=Z<br />Second step: ( a-Y)/X=X/(c-Z) ; X2= (a-Y)(c-Z) =(a-ac/(a+c))(c-ac/(a+c))<br />And following X2= a2.c2/(a+c)2 and X=ac/(a+c) =Y=Z and consequently MQNB is a square<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-61800387445326418412015-04-17T07:36:01.477-07:002015-04-17T07:36:01.477-07:00Construim perpendiculara in M pe BC si notam cu P ...Construim perpendiculara in M pe BC si notam cu P intersectia ei cu AC.Aratam ca P=Q adica MP=BM=NB=x,trENA~trCNB=>c/a=(c-x)/x=>cx=a(c-x);PQ||AB=>trCMP~trCBA=> (a-x)/a=PM/c <=><br />1-x/a=PM/c <=>c-cx/a=PM=>c-a(c-x)/a=PM=>PM=x,unde a=BC,c=AB,x=BN,=>BMPN-patrat(din problema 1109 avemBN=BM) si in final P=Q,BQ este bisectoareion radunoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-71023870662806407742015-04-17T06:24:39.461-07:002015-04-17T06:24:39.461-07:00AQ/QC = AB/BC = AB/CF = AM/MF
Thus QM//CF//AB.
S...AQ/QC = AB/BC = AB/CF = AM/MF<br />Thus QM//CF//AB. <br /><br />Similarly QN//AE//BC. <br /><br />Since BM=BN, so BMQN is square. Jacob HA (EMK2000)https://www.blogger.com/profile/17238561555526381028noreply@blogger.com