Side of Spherical Triangle is Less than 2 Right Angles
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Theorem
Let $ABC$ be a spherical triangle on a sphere $S$.
Let $AB$ be a side of $ABC$.
The length of $AB$ is less than $2$ right angles.
Proof
$A$ and $B$ are two points on a great circle $E$ of $S$ which are not both on the same diameter.
So $AB$ is not equal to $2$ right angles.
Then it is noted that both $A$ and $B$ are in the same hemisphere, from Three Points on Sphere in Same Hemisphere.
That means the distance along $E$ is less than one semicircle of $E$.
The result follows by definition of spherical angle and length of side of $AB$.
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $2$. The spherical triangle.