drawn from the origin to the vertices of the triangle (on the unit sphere). The arc subtends an angle of magnitude at the centre and therefore . Introduce a Cartesian basis with along the -axis and in the -plane making an angle with the -axis. The vector projects to in the -plane and the angle between and the -axis is . Therefore, the three vectors have components:

The scalar product in terms of the components is

Equating the two expressions for the scalar product gives

This equation can be re-arranged to give

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