IMO 1959 - Problem 6

Problem: Let $\alpha$ and $\beta$ be two planes intersenting at a line $p$. In $\alpha$ a point $A$ is given and in $\beta$ a point $C$ is given, neither of which lies on $p$. Construct $B$ in $\alpha$ and $D$ in $\beta$ such that $ABCD$ is an equilateral trapezoid, $AB\parallel CD,$ in which a circle can be inscribed.
Solution: 
Analysis. For $AB\parallel CD$ to hold evidently neither must intersect $p$ and hence constructing lines $r$ in $\alpha$ through $A$ and $s$ in $\beta$ through $C$, both being parallel to $p$, we get that $B\in r$ and $D\in s$. Hence the problem reduces to a planar problem in $\gamma$, determined by $r$ and $s$. Denote by $A'$ the foot of the perpendicular from $A$ to $s$. Since $ABCD$ is isosceles and has an incircle, it follows $AD=BC=\frac{AB+CD}{2}=A'C$. The remaining parts of the problem are now obvious.