Notes
three squares in a circle solution

Solution to the Three Squares in a Circle Puzzle

Solution by the Intersecting Chords Theorem, Lengths in a Square, and Area of a Triangle

With the points as labelled in the diagram, $A P D$ and $B P E$ are chords of the circle that intersect at $P$. Therefore, by the intersecting chords theorem, the lengths of the segments satisfy:

Now, $B P$ is the diagonal of the square with $A P$ as side length, so $B P$ is $\sqrt{2} \cdot A P$. This means that $P D$ must be $\sqrt{2} \cdot P E$. Since $P E$ is the diagonal of the square with $P F$ as side, and $P C$ is the diagonal with $P D$ as side:

Lastly, triangle $B P C$ is right-angled since $B P$ and $P C$ are diagonals of aligned squares.

Therefore, the area of triangle $B P C$ is: