Notes
circle and semi-circle in triangle solution

Solution to the Circle and Semi-Circle in Triangle Puzzle

Solution by Lengths in an Equilateral Triangle and Area Scale Factor

In the above diagram, the points $B$ and $C$ are such that $B F$ and $C D$ are both perpendicular to $A C$. As the line $A D$ cuts the equilateral triangle in half, angle $B \hat{A} F$ is $30^\circ$, so triangle $A F B$ is half an equilateral triangle. Similarly, so is triangle $A D C$. This means that the length of $A D$ is twice that of $C D$ and the length of $A F$ of twice that of $F B$. So the length of $A E$ is three times the radius of the small circle and is the same length as the radius of the large semi-circle. The area scale factor is then $9$, so the large semi-circle has area $9 \times \frac{1}{2} \times 6 = 27$.