Geometry Problem 1606: Nested Squares Area Challenge

Triangle Problem 1606: Area of Nested Squares and Segment HL

Let $\triangle ABC$ be a right triangle with the right angle at vertex $B$.

The square $\mathbf{BDEF}$ is inscribed in $\triangle ABC$, with $D \in BC$, $F \in AB$, and $E \in AC$.

The square $\mathbf{FGHJ}$ is inscribed in $\triangle AFE$, with $G \in FE$, $J \in AF$, and $H \in AE$.

The square $\mathbf{DKLM}$ is inscribed in $\triangle EDC$, with $K \in DC$, $M \in DE$, and $L \in EC$.

If the area of the square constructed on the segment $\mathbf{HL}$ is $\mathbf{100}$, compute the sum of the areas of squares $\mathbf{FGHJ}$ and $\mathbf{DKLM}$.

*Note: Solving this problem may involve considering concepts such as the similarity of nested right triangles, properties of parallel lines and the parallelogram, congruence arguments, the relationship between area, and the application of the Pythagorean theorem.*

Geometry Problem 1606: Nested Squares Area Challenge

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