tag:blogger.com,1999:blog-6933544261975483399.post5361540380805982842..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Problem 520: Triangle, Six Squares, Areas, RatioAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-6933544261975483399.post-29506364553585313002010-09-15T20:23:55.146-07:002010-09-15T20:23:55.146-07:00To Emil
Thank you for your explanation.
It is nic...To Emil<br /><br />Thank you for your explanation.<br />It is nice if you can give reference in your comment so that people can follow your logic.Peter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-35006615845442198032010-09-15T16:07:21.972-07:002010-09-15T16:07:21.972-07:00Hello Peter,
Is my last explanation enough for yo...Hello Peter,<br /><br />Is my last explanation enough for you, or not?Emil Ekkernoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-19466139892619649512010-09-13T02:21:09.596-07:002010-09-13T02:21:09.596-07:00Hello Peter,
See problem 502, and c.t.e.o's s...Hello Peter,<br /><br />See problem 502, and c.t.e.o's solution.<br />BM=AC/2, in addition, DM=median line from B (=BB' of my solution).Emil Ekkernoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-15964858194168555192010-09-12T21:29:39.832-07:002010-09-12T21:29:39.832-07:00To Emil Ekker
It is not clear to me how do you h...To Emil Ekker <br /><br />It is not clear to me how do you have "Length AA' is (side of S4)/2" in your comment. Please explainPeter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-38172730896375391902010-09-12T21:18:55.289-07:002010-09-12T21:18:55.289-07:00Let c=AB, a=BC, b=AC
S1=c^2 , S2=b^2, S3=a^2
S4=b^...Let c=AB, a=BC, b=AC<br />S1=c^2 , S2=b^2, S3=a^2<br />S4=b^2+c^2-2.b.c.Cos(obtuse A)=b^2+c^2+2.b.c.Cos(A)<br />Replace 2.b.c.Cos(A)=b^2+c^2-a^2<br />We get S4=2.b^2+2.c^2-a^2<br />Similarly S5=2.a^2+2.b^2-c^2 and S6=2.a^2+2.c^2-b^2<br />S4+S5+S6=3.a^2+3.b^2+3.a^2 and (S4+S5+S6)/(S1+S2+S3)=3 <br /><br />Peter TranPeter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-41652137527722605932010-09-12T16:43:08.280-07:002010-09-12T16:43:08.280-07:00Draw three median lines on triangle ABC, AA', ...Draw three median lines on triangle ABC, AA', BB', and CC'.<br />Length AA' is (side of S4)/2, BB'=S6 side/2, CC'=S5 side/2.<br /><br />Let AB=c, BC=a, CA=b, and AA'=d, BB'=e, CC'=f.<br />Par median theorem:<br />a^2+b^2=2*(f^2+(c/2)^2),<br />b^2+c^2=2*(d^2+(a/2)^2),<br />c^2+a^2=2*(e^2+(b/2)^2).<br />---><br />2*(a^2+b^2+c^2)=2*(d^2+e^2+f^2)+(a^2+b^2+c^2)/2<br />---><br />3*(a^2+b^2+c^2)=4*(d^2+e^2+f^2)=(2d)^2+(2e)^2+(2f)^2<br />---><br />3*(S1+S2+S3)=S4+S5+S6Emil Ekkernoreply@blogger.com