Online Geometry theorems, problems, solutions, and related topics.
Geometry ProblemLevel: Mathematics Education, High School, Honors Geometry, College.Click the figure below to see the complete problem 784.
http://img402.imageshack.us/img402/4156/problem784.pngDraw lines per attached sketchNote that P and Q are midpoints of BC’ and CB’Quadrilaterals MHEP and MFQH are cyclicIn quadrilateral MHEP, ∠MEH=∠MPH=∠PHC’In quadrilateral MFQH , ∠MFH=∠MQH=∠QHB’∆BHC’ similar to ∆ CHB’ => HC’/HB’=BC’/CB’=(.5.BC’)/(.5.CB’)=PC’/QB’So ∆PHC’ similar to ∆QHB’ …( case SAS)And ∠PHC’=∠QHB’∠MEH=∠MFH => triangle MFE is isosceles => H is the midpoint of EF
Let O be the circumcenter of triangle ABC. Also, let M₁ be the midpoint of AH. From M₁ build E₁F₁ perpendicular to OM₁OMHM₁ is a ParallelogramOM₁ ǁ MH → E₁F₁ ǁ EF M₁→ midpoint [E₁F₁] →H midpoint of [EF]
consider problems 781,782,783 and the dilatation with center B and ratio 2.