Is there an Intercept theorem (from Thales, but don’t mistake it with the Thales theorem in a circle) in hyperbolic geometry?

**Euclidean Intercept Theorem:** Let S,A,B,C,D be 5 points, such that SA, SC, AC are respectively parallel to SB, SD, BD; and SB > SA, SD > SC. Then, we have SB/SA = SD/SC = BD/AD

(see wiki illustration)

**Hyperbolic Intercept Theorem?** The concept of parallel lines is tricky here… Let’s place ourselves in the Klein model, consider a triangle $ 0xy$ , two positive real numbers $ p,q$ , and construct the Einstein midpoints $ x’=m(x,0;p,q)$ and $ y’=m(y,0;p,q)$ , defined by $ $ m(a,b;p,q)=\dfrac{p\gamma_a a+q\gamma_b b}{p\gamma_a +q\gamma_b},$ $ with $ \gamma_x=1/\sqrt{1-||x||^2}$ the Lorentz factor. The analogue of “parallel lines” here resides in that we use the same $ (p,q)$ for $ x$ and $ y$ . Then, I can prove that $ $ \dfrac{\sinh d(x’,0)}{\sinh d(x,x’)}= \dfrac{\sinh d(y’,0)}{\sinh d(y,y’)},$ $ and I can also prove using hyperbolic sine law hat \begin{equation} \sinh d(x’,y’) \leq\min \left(\dfrac{\sinh d(x’,0)}{\sinh d(x,0)}\dfrac{\sin \alpha}{\sin \alpha’},\dfrac{\sinh d(y’,0)}{\sinh d(y,0)}\dfrac{\sin \beta}{\sin \beta’} \right)\sinh d(x,y). \end{equation} and I want to prove that either $ \dfrac{\sin \alpha}{\sin \alpha’} \leq 1$ or $ \dfrac{\sin \beta}{\sin \beta’}\leq 1$ . Here $ \alpha,\beta$ are the angles at $ x$ and $ y$ in $ Oxy$ , and $ \alpha’,\beta’$ at $ x’$ and $ y’$ in $ Ox’y’$ .

Exploiting negative curvature, I could already prove that $ \alpha+\beta < \alpha’+\beta’$ , but that’s not enough…

I would also be happy with a proof that Einstein midpoint operations are contracting (in $ \sinh$ of distance), when we use the same weights $ (p,q)$ .