Notes
three congruent triangles in a semi-circle solution

Solution to the Three Congruent Triangles in a Semi-Circle Puzzle

Solution by Circle Geometry

In the diagram above, the point $O$ is on the base of the semi-circle so that $O C$ is parallel to $A B$. Since $A O$ is parallel to $B C$, the quadrilateral $A O C B$ is a parallelogram. The lengths of $A B$ and $B C$ are the same since the triangles are all congruent, so the quadrilateral is actually a rhombus. This means that the lengths of $O A$ and $O C$ are equal (and both equal to the length of $B C$). The point $O$ is therefore on the diameter and equidistant from two points on the circumference, hence is the centre of the circle. The radius is therefore $10$ and the area of the semi-circle is $50 \pi$.