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

 * Polar-triangle.png

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:
 * $A$ is a pole of $B'C'$
 * $B$ is a pole of $A'C'$.

Hence, by definition, $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.