Notes
two equilateral triangles in a semi-circle solution

Two Equilateral Triangles in a Semi-Circle

Solution by Symmetry and Properties of Equilateral Triangles

The above diagram consists of reflecting the original one in the base of the semi-circle and adding in the smaller triangles on the left. That $C E$ is the continuation of $A E$ comes from filling in all the $60^\circ$ angles at $E$. Since $D$ and $E$ are equally spaced on the diagonal, reflection about the central vertical line shows that $A B C$ is an equilateral triangle and the circle is its circumcircle. The height of the centre of the circle above $B C$ is therefore one third of the height of $A$ above $B C$, and so $E C$ is one third of $A C$. Since $E C$ has the same length as $F E$, this means that $F E$ is half of $A E$ and so the larger of the triangles has four times the area of the smaller. It therefore has area $12$.