Side of Spherical Triangle is Supplement of Angle of Polar Triangle
Theorem
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Then $A'$ is the supplement of $a$.
That is:
- $A' = \pi - a$
and it follows by symmetry that:
- $B' = \pi - b$
- $C' = \pi - c$
Proof
Let $BC$ be produced to meet $A'B'$ and $A'C'$ at $L$ and $M$ respectively.
Because $A'$ is the pole of the great circle $LBCM$, the spherical angle $A'$ equals the side of the spherical triangle $A'LM$.
That is:
- $(1): \quad \sphericalangle A' = LM$
From Spherical Triangle is Polar Triangle of its Polar Triangle, $\triangle ABC'$ is also the polar triangle of $\triangle A'B'C'$.
That is, $C$ is a pole of the great circle $A'LB'$.
Hence $CL$ is a right angle.
Similarly, $BM$ is also a right angle.
Thus we have:
\(\ds LM\) | \(=\) | \(\ds LB + BM\) | ||||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds LB + \Box\) | where $\Box$ denotes a right angle |
By definition, we have that:
- $BC = a$
\(\ds BC\) | \(=\) | \(\ds a\) | by definition of $\triangle ABC$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds LB + a\) | \(=\) | \(\ds LC\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds LB\) | \(=\) | \(\ds \Box - a\) | as $LC = \Box$ |
Then:
\(\ds \sphericalangle A'\) | \(=\) | \(\ds LM\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds LB + \Box\) | from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\Box - a} + \Box\) | from $(3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \Box} - a\) |
where $2 \Box$ is $2$ right angles, that is, $\pi$ radians.
That is, $A'$ is the supplement of $a$:
- $A' = \pi - a$
By applying the same analysis to $B'$ and $C'$, it follows similarly that:
- $B' = \pi - b$
- $C' = \pi - c$
$\blacksquare$
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $11$. Polar formulae.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polar triangle