## Task 3 - 190833

Given a square $A B C D$ with side length $a$.

A line parallel to $A B$ intersects the sides $B C$ and $A D$ at points $E$ and $F$, respectively. A line parallel to $B C$ intersects $A B$ and $E F$ at points $G$ and $H$, respectively, and a line parallel to $A B$ intersects the segments $B E$ and $G H$ at points $J$ and $K$, respectively.

a) Determine the perimeter of the rectangle $K J E H$ in terms of $a$ under the condition that the rectangles $A G H F, G B J K, K J E H$, and $F E C D$ are equal in area!

b) Determine the area of the rectangle $K J E H$ in terms of $a$ under the condition that the rectangles $A G H F, G B J K, K J E H$, and $F E C D$ are equal in perimeter!
a) By assumption, each of the four mentioned rectangles has the area

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0712.jpg?height=385&width=377&top_left_y=1478&top_left_x=311)
area \(\frac{a^{2}}{4}\). From this it follows that

\[
\begin{aligned}
& D P=\frac{a^{2}}{4} : C D=\frac{a}{4} \quad ; \quad A F=\frac{3}{4} a \\
& F H=\frac{a^{2}}{4} : A F=\frac{a^{2}}{4} : \frac{3}{4} a=\frac{a}{3} \quad ; \quad E H=\frac{2}{3} a \\
& E J=\frac{a^{2}}{4} : E H=\frac{a^{2}}{4} : \frac{2}{3} a=\frac{3}{8} a
\end{aligned}
\]

Thus, the desired perimeter is \(2 E H + 2 E J = \frac{4 a}{3} + \frac{3 a}{4} = \frac{3}{8} a\).

b) We set \(B J = x\), \(K J = y\). Since the rectangles \(G B J K\) and \(K J E H\) have equal perimeters, \(2(x + y) = 2(J E + y)\), thus \(J E = x\).

Since \(A G H F\), \(G B J K\), and \(F E C D\) have equal perimeters, the sum of the half-perimeters of \(A G H F\) and \(G B J K\) is equal to the perimeter of \(F E C D\), i.e., \(F A + A G + G B + B J = 2(C D + C E)\), which means

\[
3 x + a = 2(a + a - 2 x)
\]

From this it follows that \(x = \frac{3}{7} a\). Since \(A G H F\) and \(G B J K\) have equal perimeters, \(F A + A G = G B + B J\), which means

\[
\frac{6}{7} a + a - y = y + \frac{3}{7} a
\]

From this it follows that \(y = \frac{5}{7} a\). Thus, the desired area is \(x y = \frac{15}{49} a^{2}\).