Question

In a triangle abc there are points d ,e and f which divides line ab,bc,ca respectively into ratio 1:2 , then find area of triangle def

Answer 1

For this question, it can be solved as a oneliner by Rhodes' theorem. It states that if these lines d,e,f divide ab,bc,ca in ratios x,y,z, the ratio of the area of the triangle within and the area of the original triangle is given by (xyz-1)²/{(xy + y +1)(yz + z +1)(zx +x + 1)

In this case each of the ratios is 2( you may take the ratio as 1/2 also and you will still get the same result)-

So you answer is (2×2×2 -1)²/(2×2+2+1)(2×2+2+1)(2×2+2+1) = 7²/7³ = 1/7 of the area of the original triangle.