三角形ABC,已知DE平行BC,DF平行AC,三角形ADE面积为a,三角形DBF面积为b,求四边形DFCE面积?

解:△ADE∽△BDF
DE^2:BF^2=a:b

BF=DE√(b/a)

BC=BF+FC=BF+DE=DE(1+√(b/a)

△ADE∽△ABC
DE:BC=1:(1+√(b/a)

S△ADE:S△ABC=DE^2:BC^2=1:(1+√(b/a)^2

S△ABC=S△ADE*(1+√(b/a)^2=a(1+√(b/a)^2

S□DFCE=S△ABC-a-b
=a(1+√(b/a)^2-a-b
=a[1+2√(b/a)+b/a]-a-b
=2√(ab)

DE平行BC,DF平行AC，设四边形DFCE的面积为s，则三角形ABC的面积为
a+b+s
三角形ADE相似三角形ABC，三角形DBF相似三角形ABC
由于相似比等于面积比的平方
(AD/AB)=[a/(a+b+s)]^(1/2) (1)
(BD/AB)=[b/(a+b+s)]^(1/2) (2)
(1)+(2)
1=[a/(a+b+s)]^(1/2)+[b/(a+b+s)]^(1/2)
(a+b+s)^(1/2)=(a^(1/2)+b^(1/2)
s=2*(ab)^(1/2)

s=2*(ab)^(1/2)