Area of quadrilateral from an equilateral triangle inside a square

Question

In the image, $ABCD$ is a square of area $1$ and $\triangle DCE$ is an equilateral triangle. If $F$ and $G$ are the midpoints of $AD$ and $BC$ respectively, find the area of the quadrilateral $HJEI$.

I did this problem with a pretty tedious method. I put the figure in the plane and found the coordinates of the vertices of the quadrilateral, and then applied the Shoelace formula for the area of any polygon in the plane. With this method, I found that the answer is $\frac{54\sqrt3 - 93}{88}$.

Is there a better way to do this problem? I think there are plenty of ways but I couldn't find a better way than this.

Answer 1

Note

$$EH = PH - (PQ-EQ) = \frac14 - (1-\frac{\sqrt3}2)=\frac{2\sqrt3-3}4=a$$

and the triangles $\triangle HIE$ and $\triangle ADI$ are similar, which leads to,

$$\frac{EI}{ED-EI}=\frac{EH}{AD} \implies EI = \frac a{1+a}$$

The area of the quadrilateral $HJEI$ is twice the area of $\triangle HIE$ with $\angle IEH = 30^\circ$. Thus,

$$A = EH\cdot EI \cdot \sin30^\circ = \frac12 \frac {a^2}{1+a} =\frac12 \frac{\left(\frac{2\sqrt3-3}4\right)^2}{1+\frac{2\sqrt3-3}4}=\frac{54\sqrt3-93}{88}$$