12.12. Two planes, parallel to a given plane $\Pi$, intersect the edges of a trihedral angle at points $A, B, C$ and $A_{1}, B_{1}, C_{1}$ (points denoted by the same letters lie on the same edge). Find the geometric locus of the points of intersection of the planes $A B C_{1}$, $A B_{1} C$ and $A_{1} B C$.
12.12. The intersection of the planes $A B C_{1}$ and $A B_{1} C$ is the line $A M$, where $M$ is the point of intersection of the diagonals $B C_{i}$ in $B_{1} C$ of the face $B C C_{1} B_{1}$. The point $M$ lies on the line $l$, passing through the midpoints of the segments $B C$ and $B_{1} C_{1}$ and the vertex of the given trihedral angle (see Shrasolov, 1.22). The line $l$ is uniquely determined by the plane $P$; therefore, the plane $\Pi_{a}$, containing the line $l$ and the point $A$, is uniquely defined. The point of intersection of the line $A M$ with the plane $A_{1} B C$ belongs to the plane $\Pi_{a}$, since the entire line $A M$ belongs to this plane. Similarly, $\Pi_{a}$ constructs the plane $\Pi_{b}$; let $m$ be the line of intersection of these planes (the plane $\Pi_{c}$ also passes through the line $m$). The desired GMT consists of the points of this line lying inside the given trihedral angle.