2026/10/01

$AB$ is the diameter of semicircle, and $AB=20$. There are two squares inscribed inside the semicircle. Find the sum of the area of two squares $S_1 + S_2$.

Solve:

\(\begin{multline} \shoveleft \text{Let points }C, D \text{ to be the intersect of the squares and the semicircle}\\ \shoveleft \text{Let points }C', D' \text{ to be their reflection along }AB, E \text{ to be their shared point on }AB\\ \shoveleft \text{Let the side length of the squares to be }a, b \implies S_1=a^2, S_2=b^2\\ \shoveleft \text{Easy to see that }CE \perp DE, CE \perp C'E \implies C', E, D \text{ are collinear}\\ \shoveleft \text{Similarly, }C,E,D' \text{ are collinear} \implies CD', C'D \text{ are orthogonal chords}\\ \shoveleft \implies CE^2+ED^2+C'E^2+D'E^2=AB^2 (\href{https://en.wikipedia.org/wiki/Perpendicular#cite_note-4}{\text{According to Anonymous Theory}})\\ \shoveleft \implies 2*(\sqrt{2}a)^2+2*(\sqrt{b})^2=20^2\implies S_1+S_2=a^2+b^2=\bbox[5px, border: 1px solid black]{100} \end{multline}\)