tag:blogger.com,1999:blog-6933544261975483399.post8890177046617589825..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Problem 545: Acute Triangle, Squares, Altitudes, AreaAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6933544261975483399.post-91366196051897902472010-12-03T03:16:13.842-08:002010-12-03T03:16:13.842-08:00B, C, E, F are concyclic; AEC, AFB are secants
So ...B, C, E, F are concyclic; AEC, AFB are secants<br />So AB.AF = AC.AE. Similarly<br /> BC.BD = BA.BF <br /> CA.CE = CB.CD <br /> <br />2(AB.AF + BC.BD + AC.CE)<br />=(AB.AF + BC.BD + AC.CE)+(AB.AF + BC.BD + AC.CE)<br />=(AC.AE + BA.BF + CB.CD)+(AB.AF + BC.BD + AC.CE)<br />=(AC.AE + AC.CE)+(BF.BA+AB.AF)+(CD.CB+BC.BD)<br />=AC(AE+CE)+ AB(AF+BF)+BC(BD+CD)<br />=AC.AC + AB.AB + BC.BC <br />=a^2 + b^2 + c^2<br />=Sa + Sb + ScPravinhttps://www.blogger.com/profile/05947303919973968861noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-73363532523571639962010-12-02T13:11:16.796-08:002010-12-02T13:11:16.796-08:00Apply cosine formula in triangle ABC we have:
2.b....Apply cosine formula in triangle ABC we have:<br />2.b.c.Cos(A)=b^2+c^2-a^2<br />2.a.c.Cos(B)=a^2+c^2-b^2<br />2.a.b.Cos(C)=a^2+b^2-c^2<br /><br />add both side we get 2(b.c.Cos(A)+a.c.Cos(B)+a.b.Cos(C))=a^2+b^2+c^2<br />This will get to the result<br /><br />Peter TranPeter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-14097483265159249632010-12-02T12:14:55.859-08:002010-12-02T12:14:55.859-08:00AB² - AB∙BF = AC∙AE see P 521
AC² - AC∙AE = B...AB² - AB∙BF = AC∙AE see P 521<br />AC² - AC∙AE = BC∙DC<br />BC² - BC∙DC = AB∙BF<br />=><br />AB² + AC² + BC² = 2∙( AB∙BF + AC∙AE + BC∙DC )c .t . e. onoreply@blogger.com