Find geometric derivation of $\rho(a,b)=\frac{2|a-b|}{\sqrt{1+|a|^2}\sqrt{1+|b^2|}}$ for stereographic projection.

Question

When the complex plane is projected to the spherical surface, we can brute force the formula for the the distance between the two image points $a,b$ on the sphere

$d(a,b)=\frac{2|a-b|}{\sqrt{1+|a|^2}\sqrt{1+|b^2|}}$

In retrospect, we see the factors appearing in this distance formula represent lengths. For instance $\sqrt{1+|a|^2}$ is the distance from North Pole of the unit sphere to $a$ in the complex plane.

Can we derive this formula by pure geometric argument?