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consider an equilateral spherical triangle (living on a unit sphere) defined by the interior angle of each of its corners. How can I compute the arc length of one of its vertices to the mid-point of the opposing edge (which is called "height" in a euclidean triangle) ?

Cheers !

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You can use sine rule for half triangle. If $a$ is the side and $A$ is the angle, then:

$$ \frac{\sin \pi/2}{\sin a} = \frac{\sin A/2}{\sin a/2}=\frac{\sin A}{\sin h}. $$

Or you can use the cosine rule for half triangle: $$ \cos a = \cos h\cos \frac a2+\sin h\sin \frac a2\cos\frac\pi2=\cos h\cos\frac a2 $$

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