Area of triangle formed by points where medians are intersecting the sides

Question

I am looking for a way to find the area of triangle that is formed by points where medians are intersecting the sides in order to solve another question related to medians and triangle.

What will be the area of triangle DEF if the area of the given triangle is 'A'?

Thanks!

Answer 1

Let the position vectors of $B,C$ are $\vec b, \vec c$, then the area of the triangle ABC is $|\vec b \times \vec c|/2$. The position vectors of points F, E, D are $\vec b/2, \vec c/2, (\vec b+ \vec c)/2$. Area of the small triangle DEF is $|\vec{DE} \times \vec{DF}|=\frac{1}{8}| \vec b \times \vec c|$. So the required fraction is $1/4$.