# Generalizing Stewart’s theorem to timelike triangles in Lorentzian spaces of constant curvature

Stewart’s theorem describes the length relations of the sides of a triangle and any one of its cevians in flat space ($$n$$-dimensional Euclidean space), and also in flat spacetime ($$1 + n$$-dimensional Minkowski space).

With names assigned as in the Wikipedia figure: $$a, b, c$$ (for the lengths) of triangle sides, $$d$$ of a cevian between side $$a$$ and the opposite vertex, and $$n, m$$ for the resulting segments of side $$a$$, Stewart’s theorem is stated as

$$b^2 , m + c^2 , n = a , (d^2 + m , n ),$$

of course together with

$$a = n + m;$$

where in application to Minkowski space the squares of lengths are understood as suitably signed values of spacetime intervals $$s^2$$, and the “plain” lengths as correspondingly signed square roots $$text{Sgn}( s^2 ) , sqrt{ s^2 , text{Sgn}( s^2 ) }$$.

I’m interested in corresponding relations of (suitably generalized) lengths, or in other words: in generalizations of Stewart’s theorem, for triangles and their cevians in spaces and in (Lorentzian) spacetimes of constant curvature $$k$$; especially for their roles as model triangles in the characterization of $$text{CAT}( , k , )$$ spaces by triangle comparison.
(See also M. Kunziger, C. Sämann, “Lorentzian length spaces” (math:1711.08990), sect. 4.)

In spherical geometry the generalization is readily gotten; cmp. “Stewart theorem validity on a sphere” (mse/q/3696576). Assigning names of lengths again as above, together with $$theta$$ and $$theta^{prime}$$, respectively, for the complementary angles along side $$a$$ of cevian $$d$$, opposite side $$c$$ and opposite side $$b$$, we have

begin{align*} text{Cos}( , theta , ) , text{Sin} ! ! left( , frac{sqrt{ k } , d}{2 , pi} , right) &= frac{text{Cos} ! ! left( , frac{sqrt{ k } , c}{2 , pi} right) , – , text{Cos} ! ! left( , frac{sqrt{ k } , d}{2 , pi} right) , text{Cos} ! ! left( , frac{sqrt{ k } , m}{2 , pi} right)}{text{Sin} ! left( , frac{sqrt{ k } , m}{2 , pi} , right)_{phantom{y}}} & \ &= frac{text{Cos} ! ! left( , frac{sqrt{ k } , d}{2 , pi} right) , text{Cos} ! ! left( , frac{sqrt{ k } , n}{2 , pi} right) , – , text{Cos} ! ! left( , frac{sqrt{ k } , b}{2 , pi} right)}{text{Sin} ! ! left( , frac{sqrt{ k } , n}{2 , pi} , right)} &= -text{Cos}( , theta^{prime} , ) , text{Sin} ! ! left( , frac{sqrt{ k } , d}{2 , pi} , right) end{align*}

and therefore

$$text{Cos} ! ! left( , frac{sqrt{ k } , b}{2 , pi} right) , text{Sin} ! ! left( , frac{sqrt{ k } , m}{2 , pi} right) + text{Cos} ! ! left( , frac{sqrt{ k } , c}{2 , pi} right) , text{Sin} ! ! left( , frac{sqrt{ k } , n}{2 , pi} right) =\ text{Cos} ! ! left( , frac{sqrt{ k } , d}{2 , pi} right) , left( text{Sin} ! ! left( , frac{sqrt{ k } , n}{2 , pi} right) , text{Cos} ! ! left( , frac{sqrt{ k } , m}{2 , pi} right) + text{Cos} ! ! left( , frac{sqrt{ k } , n}{2 , pi} right) , text{Sin} ! ! left( , frac{sqrt{ k } , m}{2 , pi} right) right) = text{Cos} ! ! left( , frac{sqrt{ k } , d}{2 , pi} right) , text{Sin} ! ! left( , frac{sqrt{ k } , a}{2 , pi} right).$$

(Expanding the trigonometric functions in $$sqrt{ k } ne 0$$, terms up to second order cancel, the terms of third order sum up to being proportional to the above statement of Stewart’s theorem, and remaining terms are of order five and higher.)

My question:
What are corresponding expressions, generalizing Stewart’s theorem, for Lorentzian spaces of constant nonzero curvature ?

Note: Since generalized metric relations in Lorentzian spaces, a.k.a. spacetimes, $$mathcal S$$, are often expressed in terms of Lorentzian distance$$, lambda : mathcal S times mathcal S rightarrow { mathbb R_{(ge 0)} cup infty }$$, which are strictly zero for all but timelike related pairs of events, the relations being sought may apply only to timelike triangles, i.e. with all sides being timelike, and their timelike or lightlike cevians.