Question 137
We consider the cube $ABCD EFGH$ with side $a$ cm, $a \gt 0$ and the points $K$, $L$ on the segments $AC$, respectively $EG$ such that $CK = EL = \frac{AC}{4}$. Let $M$ be the midpoint of $AE$ and the points $N$, $P$ on the segment $HF$ such that
$$
HN = FP = \frac{(\sqrt{2} - 1)a}{2} \text{ cm.}
$$
Knowing that lines $MC$ and $LK$ intersect in $Q$, prove that $MNPQ$ is a regular tetrahedron.
$$
HN = FP = \frac{(\sqrt{2} - 1)a}{2} \text{ cm.}
$$
Knowing that lines $MC$ and $LK$ intersect in $Q$, prove that $MNPQ$ is a regular tetrahedron.