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What is the area of $HIJK$ quadrilateral, if the area of $ABC$ triangle is $70$, $BE=ED=DA$, and $BF=FG= GC$?

Let's put the area of 70 on hold for a moment. Furthermore, given the listed constraints and the implicit assumption that the problem is solvable using only those constraints, we choose any proportions for the outer triangle we like.

Let's examine the case of a right isosceles triangle with hypotenuse $BC$ and a base length of 3. It has an area of $frac{9}{2}$.

Letting the origin $(0,0)$ rest at $A$

• 
  $B$ is at $(0,3)$.


• 
  $C$ is at $(3,0)$.


• 
  $D$ is at $(0,1)$.


• 
  $E$ is at $(0,2)$.


• 
  $F$ is at $(1,2)$.


• 
  $G$ is at $(2,1)$.

This gives us the following lines

• 
  $AF$ is $y=2x$
  


• 
  $AG$ is $y=frac{1}{2}x$
  


• 
  $DC$ is $y=-frac{1}{3}x+1$
  


• 
  $EC$ is $y=-frac{2}{3}x+2$
  

Considering the four batches of two equations with two unknowns yields the coordinates of the remaining four points.

• 
  $H$, from $y=2x,;y=-frac{1}{3}x+1$ is $(frac{3}{7},frac{6}{7})$.


• 
  $I$, from $y=2x,;y=-frac{2}{3}x+2$ is $(frac{3}{4},frac{3}{2})$.


• 
  $J$, from $y=frac{1}{2}x,;y=-frac{2}{3}x+2$ is $(frac{12}{7},frac{6}{7})$.


• 
  $K$, from $y=frac{1}{2}x,;y=-frac{1}{3}x+1$ is $(frac{6}{5},frac{3}{5})$.

We can express the general form for the area of a quadrilateral based on it's vertices using vector arithmetic
$$A=frac{1}{2}vert(vec{J}-vec{H})times(vec{K}-vec{I})vert$$
We can plug this into Wolfram Alpha as

abs(Cross[([12/7,6/7]-[3/7,6/7]),([6/5,3/5]-[3/4,3/2])])*.5

Scaling that for our area 70 triangle, gives a final area of 9.

This differs from Peter Foreman's answer, so one of us must have made a typo or a false assumption. Frankly mine just looks wrong, judging from your diagram. In any case he deserves credit for advocating such a brute-force approach.

The area is $frac{741}{68}$ I'm quite sure. You can find the coordinates of $H,I,J,K$ by using the equations of each straight line and taking $A$ as the origin. I then used Wolfram: Alpha to find the area given the coordinates of each vertex.