Let's construct the projections of a cylinder, which is tangent to two given planes and the first trace (circle) of the projection axis. (How many solutions are possible?)
1. Solution. Let's construct the base circle of the cylinder, that is, draw a tangent circle between the axis and the two first traces. Once the base circle of the cylinder is established, we need to find the direction of its generators. However, the generators are parallel to lines that are parallel to both planes.

![](https://cdn.mathpix.com/cropped/2024_05_02_ab314ebff96f57813f41g-1.jpg?height=524&width=637&top_left_y=318&top_left_x=728)

One such line is the intersection of the two planes. Therefore, let's find the first and second projections of the intersection line, which will give us the direction of the first and second projections of the generators.

The problem has four possible solutions. This is natural because four tangent circles can be drawn to three intersecting lines. In this case, the three lines are the axis and the first traces.

(Tóbiás J. László.)

2. Solution. The first projection of the base circle should be inscribed in the triangle formed by the given traces and the projection axis. Therefore, four solutions are possible. The direction of the cylinder is determined by the intersection line of the planes, as this is the only line for which we can find parallels in both planes.

(Lázár Lajos, Budapest.)

The problem was also solved by: Simon Sándor.