# 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**