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

Labels:
area,
circumradius,
Nagel theorem,
orthic triangle,
semiperimeter,
triangle

Subscribe to:
Post Comments (Atom)

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)