## Tuesday, July 22, 2008

### Elearn Geometry Problem 139

See complete Problem 139
Triangle Area, Orthic Triangle, Semiperimeter, Circumradius. Level: High School, SAT Prep, College geometry

1. Trough trigonometry
AE=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
.-.

2. http://img607.imageshack.us/img607/2342/problem139.png
Connect 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

3. Nice, but for a Typo:
Area(ABC)=Area(ODBF)+Area(OEFA)+Area(ODCE)