Inscribe a triangle in a given triangle, whose sides are parallel to given directions.





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Let's consider the problem as solved and mark the vertices of the inscribed triangle in $\triangle ABC$ as $A_{1}, B_{1}$, and $C_{1}$. We draw a line parallel to $A_{1} C_{1}$ through $A$, which intersects $BC$ at $D$. Through $D$ and parallel to $C_{1} B_{1}$, and through $A$ and parallel to $A_{1} B_{1}$, we draw lines that intersect at $E$. Then,

$$
\triangle ADE \sim \triangle A_{1}B_{1}C_{1}
$$

since their sides are parallel, which means that the points $B, B_{1}$, and $E$ lie on a straight line.

The construction is now easily feasible, as after constructing $\triangle ADE$, $B_{1}$ is also known, and then constructing $\triangle A_{1}B_{1}C_{1}$ only involves drawing the necessary parallel lines.

The problem has 6 solutions.

(Kornél Riesz, Budapest.)

The problem was also solved by: Aczél F., Bartók I., Bayer B., Beck P., Bogdán G., Brachfeld S., Deutsch I., Haar A., Harsányi Z., Hirschfeld Gy., Kelemen M., Kertész F., Kertész G., König D., Krausz O., Lázár L., Pilczer P., Pintér M., Pivnyik I., Póka Gy., Raab R., Riesz M., Schmidl J., Szávay Z., Szmodics H., Tóbiás J. L.