In a triangle is there an angle splitter formula like angle bisector theorem but generalized?

Question

A segment starting from a vertex of a triangle splits that angle in 2 arbitrary parts. It also splits the opposite triangle side in 2 parts.
Can a relationship be established between the angle parts and the side parts, like the proportional relationship in the case of bisector?
Here's a picture:

For simplicity I drew the triangle as right, with one leg being twice as long as the other. The blue segment is bisector, magenta is a median and red splits the right angle in two of 30 and 60 degrees respectively.
I tried to reason something from the picture but I couldn't.

Answer 1

Let's say we have a figure as the one below. Stewart's Theorem states the following relationship between the sides of a triangle and a cevian:

$$ d^2=\frac{c^2n+b^2m}{m+n}-m\cdot n$$

$\hspace{3cm}$

For a relationship between sides and angles, you should use the law of sines and the law of cosines. Alternatively, you can try to form special triangles ($30°-60°-90°$, $45°-45°-90°$, $15°-75°-90°$, etc.) to solve specific problems.