Notes
three squares and two equilateral triangles solution

Solution to the Three Squares and Two Equilateral Triangles Puzzle

Solution by Areas of Similar Shapes, Area Dissection and Properties of Equilateral Triangles

The area scale factor from the largest square to the middle one is $\frac{8}{18} = \frac{4}{9}$, so the length scale factor is $\frac{2}{3}$. This means that the point $E$ in the above diagram is two-thirds of the way along its side, and so by symmetry are $B$ and $D$. Then adding in the points at one-third along each side and joining as in the diagram divides the larger equilateral triangle into nine smaller equilateral triangles. The central equilateral triangle has area equivalent to three of these, so has area one third of the larger equilateral triangle.

The area of an equilateral triangle is $\frac{\sqrt{3}}{4}$ times the square of its side length, so the area of the larger equilateral triangle is $\frac{\sqrt{3}}{4} \times 18 = \frac{9\sqrt{3}}{2}$ and therefore the area of the inner equilateral triangle is $\frac{3\sqrt{3}}{2}$.