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 2 of 5: Find the three intersections
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A1=−B−C,B1=−A−C,C1=−A−BA_1=-B-C,\quad B_1=-A-C,\quad C_1=-A-B
Detailed analysis

The line through AA parallel to DD0DD_0 is A+tD0A+tD_0. Requiring it to lie in plane BCDBCD makes the coefficient of AA zero, so t=−3t=-3 and A1=−B−CA_1=-B-C. Cyclically, B1=−A−CB_1=-A-C and C1=−A−BC_1=-A-B.