Geometry Problem 1614: Semicircle, Inscribed Rectangles, Area Equivalence

Geometry Problem 1614: Semicircle with inscribed rectangles ABCD and EFGH showing area equivalence of shaded regions

Problem Statement:

In a semicircle with diameter $AB$, let $ABCD$ be a rectangle such that $AD = AB/5$ and the side $CD$ intersects the arc of the semicircle. A second rectangle $EFGH$ is constructed such that vertices $E$ and $G$ lie on the arc, $F$ is a point exterior to the semicircle, and the side $EF$ intersects the arc at point $M$. If $EM = MF = FG$, and the sides $EH$ and $GH$ intersect $DC$ at points $J$ and $K$ respectively, prove that: \[ \text{Area}(ABCKHJD) = \text{Area}(EFGKJ) \]

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Understanding the Problem

The "golden key" to this proof lies in identifying the relationship between the ratio $EM=MF=FG$ and the circle's arc, which leads to the discovery that $\angle FMG = 45^\circ$. By linking this angular insight with the Pythagorean theorem, you can prove that both parent rectangles share an identical area of $4R^2/5$, making the equivalence of the shaded sub-regions a matter of geometric balance.