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 5 of 5: Answer the general question
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det⁡M(x,y,z)=−(x+y+z)−2=−3\det M(x,y,z)=-(x+y+z)-2=-3
Detailed analysis

For an arbitrary interior point D0=xA+yB+zCD_0=xA+yB+zC with x+y+z=1x+y+z=1 and x,y,z>0x,y,z>0, the same affine calculation gives A1=−(yB+zC)/xA_1=-(yB+zC)/x and cyclic analogues. Expanding the resulting coefficient determinant gives det⁡M(x,y,z)=−(x+y+z)−2=−3\det M(x,y,z)=-(x+y+z)-2=-3, so the volume ratio is again 33. Thus the result remains true for every point inside △ABC\triangle ABC.