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 1 of 5: Place the centroid
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D0=A+B+C3D_0=\frac{A+B+C}{3}
A tetrahedron illustrating the four vertices and its centroid construction.
Three-dimensional tetrahedron used to orient ABCD and the centroid D0.
Detailed analysis

Use affine vectors with DD as origin. Since D0D_0 is the centroid, its vector is (A+B+C)/3(A+B+C)/3.