18. (ITA 2) If an acute-angled triangle $A B C$ is given, construct an equilateral triangle $A^{\prime} B^{\prime} C^{\prime}$ in space such that lines $A A^{\prime}, B B^{\prime}, C C^{\prime}$ pass through a given point.
18. The required construction is not feasible. In fact, let us consider the special case $\angle B O C=135^{\circ}, \angle A O C=120^{\circ}, \angle A O B=90^{\circ}$, where $A A^{\prime} \cap B B^{\prime} \cap$ $C C^{\prime}=\{O\}$. Denoting $O A^{\prime}, O B^{\prime}, O C^{\prime}$ by $a, b, c$ respectively we obtain the system of equations $a^{2}+b^{2}=a^{2}+c^{2}+a c=b^{2}+c^{2}+\sqrt{2} b c$. Assuming w.lo.g. $c=1$ we easily obtain $a^{3}-a^{2}-a-1=0$, which is an irreducible equation of third degree. By a known theorem, its solution a is not constructible by ruler and compass.