Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube

Euclid's Formulation

If the sides of the opposite planes of a cube are cut in half, and planes are produced through the pieces, then the common section of the (latter) planes and the diameter of the cube cut one another in half.

For let the opposite planes $CF$ and $AH$ of the cube $AF$ have been cut in half at the points $K$, $L$, $M$, $N$, $O$, $Q$, $P$, and $R$.
And let the planes $KN$ and $OR$ have been produced through the pieces.
And let $US$ be the common section of the planes, and $DG$ the diameter of cube $AF$.
I say that $UT$ is equal to $TS$, and $DT$ to $TG$.