See complete Problem 139
Triangle Area, Orthic Triangle, Semiperimeter, Circumradius. Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Tuesday, July 22, 2008
Elearn Geometry Problem 139
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Trough trigonometry
ReplyDeleteAE=c cosA; AF=b cosA
in triangle AFE with the law of cosine:
FE²=AF²+AE²-2AF.AEcosA
=cos²A.(b²+c²*2bc cosA)=a²cos²A
then
FE=a cosA= 2RsinAcosA=Rsin(2A)
the same way ED=R sin(2C); FD= Rsin(2B)
p= R(sin2A+sin2B+sin2C).(0.5)
pR= 0.5R²(sin2A+sin2B+sin2C)
but 0.5R²sin2A is the area of isoscele triangle
OBC; OB=OC=R and ang(BOC)=2ang(BAC)=2A
S=(ABC)=(AOB)+(BOC)+(COA)=pR
.-.
http://img607.imageshack.us/img607/2342/problem139.png
ReplyDeleteConnect OA, OB, OD, OF and OE
Area(ABC)=Area(ODBF)+Area(OEFA)+Area(OACE)
Each quadrilateral have diagonals perpendicular to each other ( see Problem 138)
And area of each quadrilateral = ½ * diagonal1 *diagonal2
So Area(ABC)=1/2*OB.FD+1/2*OA.FE+1/2*OC.ED
=1/2*R*(FD+FE+ED)= p.R
Peter Tran
Nice, but for a Typo:
ReplyDeleteArea(ABC)=Area(ODBF)+Area(OEFA)+Area(ODCE)