Given 4 lines: $t_{1}, t_{2}, t_{3}$ and $e$. Construct a parabola that is tangent to the first three lines, and whose focus lies on the line $e$.
In the solution to problem 686 III. (see page 18 in the January 1956 issue), we proved that the circle circumscribed around the triangle formed by three tangents of a parabola passes through the focus. By constructing this circle, its intersection points with the line \( e \) give the focus. As is known, the reflections of the focus over the tangents lie on the directrix.

Accordingly, depending on whether the circle intersects the line \( e \) at two different points, touches it, or does not intersect it, the number of solutions is \( 2, 1, 0 \). Naturally, there is no solution if the circumscribed circle does not exist, that is, if two of the tangents are parallel, or if the three tangents intersect at a single point.

Leipnik István (Bp., I., Toldy F. g. III. o. t.)