Construction of Solid Angle from Three Plane Angles any Two of which are Greater than Other Angle/Lemma

Proof

 * Euclid-XI-23b.png

Let the straight lines $AB$ and $LO$ be set out.

Le $AB > LO$.

Let the semicircle $ACB$ be described on $AB$.


 * Euclid-XI-23-Lemma.png

Using :
 * Let $AC = LO$ be fitted into the semicircle $ACB$.

Let $BC$ be joined.

From :
 * $\angle ACB$ is a right angle.

Therefore from :
 * $AB^2 = AC^2 + CB^2$

Hence:
 * $AB^2 - AC^2 = CB^2$

But $AC = LO$.

Therefore:
 * $AB^2 - LO^2 = CB^2$

So if we then cut of $OR = BC$:
 * $AB^2 - LO^2 = OR^2$