Ta có:
\(\begin{array}{l}\left. \begin{array}{l}M \in AB \subset \left( {ABB'A'} \right)\\M \in \left( {OMN} \right)\end{array} \right\} \Rightarrow M \in \left( {OMN} \right) \cap \left( {ABB'A'} \right)\\\left. \begin{array}{l}N \in A'B' \subset \left( {ABB'A'} \right)\\N \in \left( {OMN} \right)\end{array} \right\} \Rightarrow N \in \left( {OMN} \right) \cap \left( {ABB'A'} \right)\\ \Rightarrow \left( {OMN} \right) \cap \left( {ABB'A'} \right) = MN\end{array}\)
\(M\) là trung điểm của \(AB\)
\(N\) là trung điểm của \(A'B'\)
\( \Rightarrow MN\) là đường trung bình của hình bình hành \(ABB'A'\)
\( \Rightarrow MN\parallel AA'\parallel BB'\parallel CC'\parallel DD'\)
\(\left. \begin{array}{l}O \in \left( {OMN} \right) \cap \left( {C{\rm{DD'C'}}} \right)\\MN\parallel C{\rm{D}}\\MN \subset \left( {OMN} \right)\\C{\rm{D}} \subset \left( {C{\rm{DD'C'}}} \right)\end{array} \right\}\)
\( \Rightarrow \)Giao tuyến của hai mặt phẳng \(\left( {OMN} \right)\) và \(\left( {C{\rm{DD'C'}}} \right)\) là đường thẳng \(d\) đi qua \(O\), song song với \(MN\) và \(C{\rm{D}}\).
Gọi \(P = d \cap C'D',Q = d \cap CD \Rightarrow \left( {OMN} \right) \cap \left( {C{\rm{DD'C'}}} \right) = PQ\)
\(\begin{array}{l}\left. \begin{array}{l}M \in AB \subset \left( {ABC{\rm{D}}} \right)\\M \in \left( {OMN} \right)\end{array} \right\} \Rightarrow M \in \left( {OMN} \right) \cap \left( {ABC{\rm{D}}} \right)\\\left. \begin{array}{l}Q \in C{\rm{D}} \subset \left( {ABC{\rm{D}}} \right)\\Q \in d \subset \left( {OMN} \right)\end{array} \right\} \Rightarrow Q \in \left( {OMN} \right) \cap \left( {ABC{\rm{D}}} \right)\\ \Rightarrow \left( {OMN} \right) \cap \left( {ABC{\rm{D}}} \right) = MQ\end{array}\)
\(\begin{array}{l}\left. \begin{array}{l}N \in A'B' \subset \left( {A'B'C'{\rm{D'}}} \right)\\N \in \left( {OMN} \right)\end{array} \right\} \Rightarrow N \in \left( {OMN} \right) \cap \left( {A'B'C'{\rm{D'}}} \right)\\\left. \begin{array}{l}P \in C'{\rm{D'}} \subset \left( {A'B'C'{\rm{D'}}} \right)\\P \in d \subset \left( {OMN} \right)\end{array} \right\} \Rightarrow P \in \left( {OMN} \right) \cap \left( {A'B'C'{\rm{D'}}} \right)\\ \Rightarrow \left( {OMN} \right) \cap \left( {A'B'C'{\rm{D'}}} \right) = NP\end{array}\)
Gọi \(E = MQ \cap BC,F = MQ \cap AD,G = NP \cap B'C',H = NP \cap A'D'\)
\(\begin{array}{l}\left. \begin{array}{l}E \in BC \subset \left( {BCC'B'} \right)\\E \in MQ \subset \left( {OMN} \right)\end{array} \right\} \Rightarrow E \in \left( {OMN} \right) \cap \left( {BCC'B'} \right)\\\left. \begin{array}{l}G \in B'C' \subset \left( {BCC'B'} \right)\\G \in NP \subset \left( {OMN} \right)\end{array} \right\} \Rightarrow G \in \left( {OMN} \right) \cap \left( {BCC'B'} \right)\\ \Rightarrow \left( {OMN} \right) \cap \left( {BCC'B'} \right) = EG\end{array}\)
\(\begin{array}{l}\left. \begin{array}{l}F \in A{\rm{D}} \subset \left( {A{\rm{DD'A'}}} \right)\\F \in MQ \subset \left( {OMN} \right)\end{array} \right\} \Rightarrow F \in \left( {OMN} \right) \cap \left( {A{\rm{DD'A'}}} \right)\\\left. \begin{array}{l}H \in A'D' \subset \left( {A{\rm{DD'A'}}} \right)\\H \in NP \subset \left( {OMN} \right)\end{array} \right\} \Rightarrow H \in \left( {OMN} \right) \cap \left( {A{\rm{DD'A'}}} \right)\\ \Rightarrow \left( {OMN} \right) \cap \left( {A{\rm{DD'A'}}} \right) = FH\end{array}\)