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Let $ABCD$ be a regular tetrahedron.  Let $E$, $F$, $G,$ $H$ be the centers of faces $BCD$, $ACD$, $ABD$, $ABC$, respectively.  The volume of pyramid $DEFG$ is $18.$  Find the volume of pyramid $EFGH$

 Dec 26, 2023
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Note that $DEFG$ is similar to $ABCD$ with ratio $\frac12$.

 

The ratio of their volumes is $(\frac12)^3=\frac18$, so the volume of $ABCD$ is $8(18)=144$

 

Using the fact that $ABCD$ is a regular tetrahedron with side length $2$, we find that $[ABC]=12\sqrt{2}$, and so


\[[EFGH]=[ABCD]-4[ABC]=144-4\cdot12\sqrt{2}=\boxed{144-48\sqrt{2}}.\]

 Dec 26, 2023

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