Ming's Mark RA3 Blue/Beige 9' x 12' Classical Mat by Ming's Mark

here i've got a nice elementary geometry problem so we've got an equilateral triangle with side length four furthermore we have some line segments emanating from the vertices of this equilateral triangle which i've drawn in orange they intersect in a way that we create another equilateral triangle that is inside of our larger equilateral triangle and they intersect in a way so that all of these four sub-triangles have the same area so the area of this green equilateral triangle in the middle is equal to the area of this red triangle and the area of this blue triangle and the area of the purple triangle so we have four equal area triangles building up this other triangle and our goal is to find x where x is the length of this line segment from this vertex here to this vertex here okay so let's see what we can do about this first off i want to calculate the area of the entire triangle wha