\section*{Problem \(5-011135=011234\)}

Given a line segment \(A B=a=6 \mathrm{~cm} . M\) is the midpoint of the segment.

Draw a semicircle over \(A B\) with radius \(A M\) centered at \(M\)! Bisect \(A M\) and \(M B\) and draw two semicircles with radius \(\frac{A M}{2}\) over both segments, which lie within the larger semicircle! Construct the center of the circle that touches the large semicircle from the inside and the two smaller semicircles from the outside!

The construction must be justified!
}

\section*{I. Analysis:}

The point of contact of the semicircle over \(A M\) and the semicircle over \(M B\) with the circle \(k\) to be constructed is \(K\) and \(L\) respectively, and the point of contact of the semicircle over \(A B\) with \(k\) is \(N\). The midpoints of \(A M\) and \(B M\) are denoted by \(P\) and \(Q\) respectively.

Since the tangents of \(k\) and the semicircle over \(A M\) at \(K\) are identical and the tangents are perpendicular to the radii, the line segment between the center \(R\) of circle \(k\) and \(P\) passes through \(K\).

Similarly, \(L\) lies on \(R Q\). Since \(R P = \frac{A M}{2} + R K = R Q\), it follows that \(\angle P M R = 90^\circ\) and by the Pythagorean theorem,

\[
\left(\frac{A M}{2}\right)^2 + (A M - R N)^2 = \left(\frac{A M}{2} + R K\right)^2
\]

From \(R N = R K\), we get

\[
\frac{1}{4} A M^2 + A M^2 - 2 A M \cdot R N + R N^2 = \frac{1}{4} A M^2 + A M \cdot R N + R N^2
\]

Thus, \(A M^2 = 3 A M \cdot R N\), i.e.,

\[
R N = \frac{1}{3} A M = \frac{a}{6} = 1 \text{ cm} \quad \text{and} \quad M R = M T - R N = 2 \text{ cm}
\]

II. Construction Description:

(1) Construct the perpendicular bisector of \(A B\) and denote the intersection of this perpendicular bisector and the semicircle over \(A B\) by \(N\).

(2) Now construct a point \(S\) on the extension of \(A M\) beyond \(A\) such that \(S M = 3 A M\).

Construct the perpendicular to \(S M\) at \(A\), and denote the intersection of this perpendicular and \(S N\) by \(T\). Then, by the intercept theorem, \(T N = \frac{1}{3} S N\).

(3) Construct the perpendicular from \(T\) to \(M N\) and denote the foot of the perpendicular by \(R\). By the intercept theorem, \(N R = \frac{T N}{S N} M N = 1 \text{ cm}\), i.e., \(R\) is the center of the circle \(k\) that touches the smaller semicircles externally and the larger semicircle internally, as derived from the above analysis.

(4) Draw a circle around \(R\) with radius \(R N\).

III. Construction:

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1684.jpg?height=403&width=1240&top_left_y=1095&top_left_x=408)