Question 50
Let $A_{1} A_{2} A_{3} A_{4}$, $B_{1} B_{2} B_{3} B_{4}$, and $C_{1} C_{2} C_{3} C_{4}$ be three regular tetrahedra in 3-dimensional space, no two of which are congruent. Suppose that, for each $i \in \{1,2,3,4\}$, $C_{i}$ is the midpoint of the line segment $A_{i} B_{i}$. Determine whether the four lines $A_{1} B_{1}$, $A_{2} B_{2}$, $A_{3} B_{3}$, and $A_{4} B_{4}$ must concur.