Spherical Triangle is Polar Triangle of its 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 $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.
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 $ALM$.
By construction we have that $B'$ is the pole of $AC$.
Thus the length of the arc of the great circle from $B$ to any point on $AC$ is a right angle.
Similarly, the length of the arc of the great circle from $A'$ to any point on $BC$ is also a right angle.
Hence:
- the length of the great circle arc $CA'$ is a right angle
- the length of the great circle arc $CB'$ is a right angle
and it follows by definition that $C$ is a pole of $A'B'$.
In the same way:
Hence, by definition, $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.
$\blacksquare$
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