Question 4
A regular tetrahedron $ABCD$ has edge $a$. On $AB$ choose $E,E'$ with $AE=a/6$, $AE'=5a/6$; on $AC$ choose $F,F'$ with $AF=a/4$, $AF'=3a/4$; on $AD$ choose $G,G'$ with $AG=a/3$, $AG'=2a/3$.
(a) Determine the intersection lines of the plane $BCD$ with the planes $EFG$ and $E'F'G'$. Study their relative position with respect to triangle $BCD$. State a general theorem.
(b) Calculate the volume of $EFGE'F'G'$ in terms of $a$, and calculate the angles that “the two planes $EFG$” make with $AB,AC,AD$. [The quoted wording is as printed; only $EFG$ is named.]
(a) Determine the intersection lines of the plane $BCD$ with the planes $EFG$ and $E'F'G'$. Study their relative position with respect to triangle $BCD$. State a general theorem.
(b) Calculate the volume of $EFGE'F'G'$ in terms of $a$, and calculate the angles that “the two planes $EFG$” make with $AB,AC,AD$. [The quoted wording is as printed; only $EFG$ is named.]