Let ABC be any triangle. Let D be a point on side AB and let E be a point on side AC. Draw lines CD and BE and call their intersection F. Triangle ABC is then divided into three smaller triangles BDF, CEF, BCF and a quadrilateral ADFE.
Let the area of BDF equal r, the area of CEF equal q, and the area of BCF equal p. Express the area of the whole triangle ABC in terms of p, q, and r.
Let [LMN] denote the area of triangle LMN.
[ABC] = [ABF] + [BCF] + [CAF]
[ABF] [BCF] [CAF]
1 =  +  + 
[ABC] [ABC] [ABC]
[BDF] [BCF] [CEF]
1 =  +  + 
[BDC] [ABC] [CEB]
[BDF] [CEF]
[ABC] = *[ABC] + [BCF] + *[ABC]
[BDC] [CEB]
r q
[ABC] = *[ABC] + p + *[ABC]
p+r p+q
p*(p+q)*(p+r)
[ABC] = 
p*p  q*r

Posted by Bractals
on 20081201 13:19:45 