MathLabs

Problem 6

In tetrahedron ABCDABCD, vertex DD is joined to D0D_0, the centroid of △ABC\triangle ABC. Through A,B,CA,B,C draw lines parallel to DD0DD_0, meeting the planes BCD,CAD,ABDBCD,CAD,ABD at A1,B1,C1A_1,B_1,C_1, respectively. Prove that the volume of ABCDABCD is one third the volume of A1B1C1D0A_1B_1C_1D_0. Is the result true if D0D_0 is selected anywhere within △ABC\triangle ABC?
Step 3 of 5: Form the volume matrix
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M=(−1/3−4/3−4/3−4/3−1/3−4/3−4/3−4/3−1/3)M=\begin{pmatrix}-1/3&-4/3&-4/3\\-4/3&-1/3&-4/3\\-4/3&-4/3&-1/3\end{pmatrix}
Detailed analysis

Subtract D0D_0 from A1,B1,C1A_1,B_1,C_1 and express the resulting vectors in the basis A,B,CA,B,C. Their coefficient matrix is MM as displayed, while the original tetrahedron uses the identity matrix.