\section*{Problem 1 - 291041}

Given are three lines \(g, h, j\) in a plane; no two of these lines are parallel to each other; no point in the plane lies on all three lines. Furthermore, a length \(a\) is given.

For each such specification of \(g, h, j, a\), the number of all circles \(c\) that satisfy the following conditions is sought:

(1) The circle \(c\) intersects each of the lines \(g, h, j\) at two points \(G_{1}, G_{2}\) respectively, \(H_{1}, H_{2}\) respectively, \(J_{1}, J_{2}\).

(2) It holds that \(G_{1} G_{2}=H_{1} H_{2}=J_{1} J_{2}=a\).
}

I. If a circle \(c\) satisfies conditions (1) and (2), then: If \(M\) is the center and \(r\) is the radius of \(c\), then \(M\) has the distance

\[
s=\sqrt{r^{2}-\left(\frac{a}{2}\right)^{2}}
\]

from each of the lines \(g, h, j\). This follows from the theorem that the perpendicular from \(M\) to a chord bisects the chord, and from the Pythagorean theorem.

By assumption, \(g, h, j\) form the sides and their extensions of a triangle \(D\). Thus, \(M\) is one of the four points that have equal distances from the sides (including their extensions) of \(D\); i.e., \(M\) is the incenter or one of the three excenters of \(D\), and the length \(s\) mentioned in (3) is the inradius or the radius of the respective excircle. Furthermore, it follows from (3) that \(c\) has the radius:

\[
r=\sqrt{s^{2}+\left(\frac{a}{2}\right)^{2}}
\]

II. Conversely, it follows: If \(M\) is the center and \(s\) is the radius of the incircle or one of the excircles of \(D\), and if \(r\) is the length formed according to (4), then the circle \(c\) constructed around \(M\) with radius \(r\) intersects each of the three lines \(g, h, j\) in a chord whose half-length is \(\sqrt{r^{2}-s^{2}}=\frac{a}{2}\); i.e., then \(c\) satisfies conditions (1) and (2).

Thus, it is proven: For each of the given specifications of \(g, h, j, a\) in the problem, there are exactly four circles \(C\) that satisfy (1) and (2).