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

## Theorem

In the words of Euclid:

*If three straight lines be proportional, the parallelepipedal solid formed out of the three is equal to the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.*

(*The Elements*: Book $\text{XI}$: Proposition $36$)

## Proof

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:

- $A : B = B : C$

while:

- $A = LM$
- $B = LO = ED = EF$
- $C = LN$

Therefore:

- $LM = EF = DE : LN$

Thus the sides about the equal angles $\angle NLM, \angle DEF$ are reciprocally proportional.

Therefore from Proposition $14$ of Book $\text{VI} $: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional:

- the parallelogram $MN$ equals the parallelogram $DF$.

It follows from Porism to Proposition $35$ of Book $\text{VI} $: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles:

- the perpendiculars from $G$ and $O$ to the planes containing $\angle NLM$ and $\angle DEF$ are equal.

Therefore the parallelepipeds $LH$ and $EK$ are of equal height.

- $LH = EK$.

We have that:

- $LH$ is the parallelepiped formed out of $A, B, C$
- $EK$ is the parallelepiped formed out of $B$.

Hence the result.

$\blacksquare$

## Historical Note

This proof is Proposition $36$ of Book $\text{XI}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions