Areas of spherical and hyperbolic triangles in terms of their midpoints

Abstract

Let $M$ be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If $\Delta(abc)$ is a geodesic triangle on $M$ with corners at $a,b,c\in M$, we denote by $\alpha, \beta, \gamma\in M$ the midpoints of their sides. If $\Omega$ denotes the oriented area of this triangle on $M$, it satisfies the relations: $$ \sin(\Omega/2) = \frac{\Det_3(abc)}{\sqrt{2(1 + \scalar{a}b)(1 + \scalar{b}c)(1 + \scalar{c}a)}} = \Det_3(\alpha\beta\gamma) \, $$ where $\scalar{\}{\}$ denotes the Euclidean scalar product for $M=\SS^2$ and the Lorentzian scalar product for $M=\HH^2$. On the hyperbolic plane one should always take the solution with $\modu{\Omega/2}<\pi/2$. On the sphere, singular cases excepted, a straightforward procedure tells us which solution of this equation is the correct one.