\section*{Problem 4 - 031014}

Given a square with side length \(a\). Five equal-sized circles are to be drawn in this square such that one circle is in the center and the other four touch both this circle and two adjacent sides of the square.

a) Express the radius of these circles in terms of \(a\)! b) Construct the circles using only a compass and a straightedge (provide construction steps)!
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a) The diagonal of the quadrilateral has the length \(a \sqrt{2}\); and it can also be expressed in terms of \(r\), specifically as: \(r \sqrt{2} + 4r + r \sqrt{2}\).

Setting the two terms equal leads to \(r = \frac{a}{2}(\sqrt{2} - 1)\).

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1198.jpg?height=414&width=445&top_left_y=244&top_left_x=811)

b) Draw the square \(ABCD\) and its diagonals; their intersection point is \(M\). From one of the vertices, measure \(\frac{a}{2}\) along the diagonal and mark the resulting point on the diagonal as \(E\). Since \(\frac{a}{2} = r + r \sqrt{2}\), it follows that the desired radius \(r = EM\).

Adapted from [5]