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 4 of 5: Compare the volumes
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∣det⁡M∣=3|\det M|=3
Detailed analysis

The eigenvalues of (144414441)\begin{pmatrix}1&4&4\\4&1&4\\4&4&1\end{pmatrix} are 9,−3,−39,-3,-3, so ∣det⁡M∣=81/27=3|\det M|=81/27=3. Therefore VA1B1C1D0=3VABCDV_{A_1B_1C_1D_0}=3V_{ABCD}.