tag:blogger.com,1999:blog-6933544261975483399.post5710892007453135076..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Problem 784: Triangle, Orthocenter, Altitude, Midpoint of a side, PerpendicularAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6933544261975483399.post-51698968479001528662021-05-30T19:04:16.041-07:002021-05-30T19:04:16.041-07:00Why is M_1 the midpoint of E_1F_1?Why is M_1 the midpoint of E_1F_1?math_learnerhttps://www.blogger.com/profile/16314682371358089936noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-23112016034893031022012-07-18T22:37:57.744-07:002012-07-18T22:37:57.744-07:00consider problems 781,782,783 and the dilatation w...consider problems 781,782,783 and the dilatation with center B and ratio 2.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-86801675044747836162012-07-17T13:57:37.460-07:002012-07-17T13:57:37.460-07:00Let O be the circumcenter of triangle ABC. Also, ...Let O be the circumcenter of triangle ABC. Also, let M₁ be the midpoint of AH. From M₁ build E₁F₁ perpendicular to OM₁<br />OMHM₁ is a Parallelogram<br />OM₁ ǁ MH → E₁F₁ ǁ EF <br />M₁→ midpoint [E₁F₁] →H midpoint of [EF]Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-37797367813372857612012-07-17T09:43:08.848-07:002012-07-17T09:43:08.848-07:00http://img402.imageshack.us/img402/4156/problem784...http://img402.imageshack.us/img402/4156/problem784.png<br /><br />Draw lines per attached sketch<br />Note that P and Q are midpoints of BC’ and CB’<br />Quadrilaterals MHEP and MFQH are cyclic<br />In quadrilateral MHEP, ∠MEH=∠MPH=∠PHC’<br />In quadrilateral MFQH , ∠MFH=∠MQH=∠QHB’<br />∆BHC’ similar to ∆ CHB’ => HC’/HB’=BC’/CB’=(.5.BC’)/(.5.CB’)=PC’/QB’<br />So ∆PHC’ similar to ∆QHB’ …( case SAS)<br />And ∠PHC’=∠QHB’<br />∠MEH=∠MFH => triangle MFE is isosceles => H is the midpoint of EFPeter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.com