tag:blogger.com,1999:blog-6933544261975483399.post7099670097082895436..comments2024-10-13T09:45:37.126-07:00Comments on GoGeometry.com (Problem Solutions): Problem 611: Altitude of a Triangle, Perpendicular, Collinear Points, ParallelAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-6933544261975483399.post-19497538973212544802012-07-15T19:37:03.115-07:002012-07-15T19:37:03.115-07:00http://img840.imageshack.us/img840/7563/problem611...http://img840.imageshack.us/img840/7563/problem611.png<br /><br />Draw circle diameter BD ( see attached sketch)<br />From B draw a line parallel to AC . This line cut DE and DF at M and H<br />AH cut BD at G and MC cut BD at G’ .<br />By the properties of right triangles we have BH.DC=BD^2= BM.AD<br />So BM/DC=BH/AD<br />∆MBG similar to ∆ CDG => BG’/DG’= BM/DC<br />∆HBG similar to ∆ADG => BG/DG= BH/AD<br />From above expressions we have BG’/DG’= BG/DG => G coincide to G’<br />M, B and H are collinear and BH // ACPeter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-46592371976292190722011-06-05T22:36:19.462-07:002011-06-05T22:36:19.462-07:00Analytic Proof:
Let D = (o,o), A = (a,0), C = (c,0...Analytic Proof:<br />Let D = (o,o), A = (a,0), C = (c,0), B = (0,b) <br />The equations of AB, DEM are respectively<br />bx + ay = ab and ax-by = 0<br />Solving, we have<br />E = [ab²/(a² + b²) , a²b/(a² + b²)]<br />Similarly,<br />F = [b²c/ (b² + c²), bc²/ (b² + c²)]<br />It can be verified that <br />the equation of EF is <br />b (a + c)x – (b² - ac)y = abc<br />Hence <br />G = [0, abc/ (ac - b²)] and <br />so the equation of CG is <br />abx + (ac - b²)y =abc<br />Solving it with the equation <br />ax-by = 0 of DE<br />we obtain (eliminating x)<br />y-coord of M = b = y-coord of B<br />M and B are at the same height over AC<br />Hence BM is parallel to AC<br />Similarly BH is parallel to AC<br />Thus M, B, H are collinearPravinhttps://www.blogger.com/profile/05947303919973968861noreply@blogger.com