Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it formed on Mean

Proof

 * Euclid-XI-36.png

Let $A, B, C$ be three straight lines in proportion:
 * $A : B = B : C$

Let a solid angle be constructed at $E$ contained by the plane angles $\angle DEG, \angle GEF, \angle FED$.

Let each of the straight lines $DE, GE, EF$ be made equal to $B$.

Let the parallelepiped $EK$ be completed.

Let the straight line $LM$ be made equal to $A$.

Let a solid angle be constructed at $L$ contained by the plane angles $\angle NLO, \angle OLM, \angle MLN$.

Let the straight line $LO$ be made equal to $B$.

Let the straight line $LN$ be made equal to $C$.

We have that $A : B = B : C$.