EDIT1:
On a spherical surface radius $R$ a geodesic triangle is drawn:
Let a,b, and c be the lengths of the sides of the geodesic triangle. Let d be the geodesic arc length of a cevian to the side of length a. The cevian divides the side of length a into two geodesic arc segments of length m and n, with m adjacent to c and n adjacent to b.
Would a * spherical Stewart's theorem*
$${\displaystyle b^{2}m+c^{2}n=a(d^{2}+mn)}? $$
be still valid? I do not think so.
If not, what changes in the angles of spherical triangle be incorporated into a new relation?
The radius of sphere is thrice the circum-diameter of a plane triangle with these dimensions.
We have $\sin a \cos d = \sin m \cos b + \sin n \cos c$ for a spherical triangle. And by third-order approximation, we can recover the original Stewart theorem $a^3 + 3ad^2 = m^3 + 3mb^2 + n^3 + 3nc^2$.