Notes
two isosceles triangles solution

Two Isosceles Triangles

Solution by Properties of Isosceles Triangles and Equilateral Triangles

As both isosceles triangles have the same base and area, they must have the same height. So in the above diagram, $F B$ and $E D$ have the same length. Since triangle $F A C$ is an isosceles right-angled triangle and $B$ is the midpoint of $A C$, triangle $F B C$ is also isosceles and right-angled and so the length of $F B$ is the same as that of $B C$ and so half of that of $A C$. This means that $E D$ has half the length of $C E$, so triangle $C E D$ is half an equilateral triangle. Angle $D \hat{C} E$ is therefore $30^\circ$ so angle $E \hat{C} A$ is $150^\circ$, and finally angle $A \hat{E} C$ is $15^\circ$.