See complete Problem 219 at:
gogeometry.com/problem/p219_rhombus_perpendicular_arithmetic_mean.htm
Level: High School, SAT Prep, College geometry
Sunday, January 4, 2009
Elearn Geometry Problem 219: Rhombus, perpendiculars, arithmetic mean
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clearly HM=HO=HG=x and OF=NF/2, so
ReplyDeletea+x=(b+c+2x)/2
a=(b+c)/2
ABCD rhombus => GHO isoscele in H => OH=HG=HM since GOM rectangle in O => OH=GM/2
ReplyDeleteAlso, ABCD rhombus => FN distance between AB and CD is the same as that between AD and BC which is 2.OE
So a=OE-OH=(FN-GM)/2=(FG+MN)/2=(b+c)/2 QED