A short note on the Median ratio property in Spherical Geometry

We know that in a plane triangle, the median is divided by the centroid in the ratio $2:1$. In this post, we address the question of how this relation generalizes in Spherical geometry.

Consider a spherical triangle $ABC$ with $D$, $E$ and $F$ being the midpoints of $BC$, $CA$ and $AB$. We know from the spherical analogy of Ceva's theorem, that the three medians $AD$, $BE$ and $CF$ meet at a common point, say $G$.

Using the volume and sine-of-side relation, we know that $|ABG|=|BCG|=|CAG|$.

Then, using our results from this post, we have,

$\displaystyle \frac{\sin AG}{\sin GD}=\frac{\sin BC}{\sin BD}$

Using the fact that $D$ is the midpoint of the arc $BC$,

$\displaystyle \frac{\sin AG}{\sin GD}=\frac{\sin BC}{\sin BD}=\frac{2\sin BD \cos BD}{\sin BD}=2\cos(BC/2)$

This shows that, unlike the planar case, the median ratio is dependent on the side on which the median rests ($BC$ in this case). However, by the same relation, we see that it is independent of the vertex of the median. That is, no matter the position of $A$ and as long as the $BC$ is fixed, the ratio remains invariant.

Hope you enjoyed the post.

Yours Aye

Me