Does the median make angles in the same proportion as the sides?

Question

Till I remember I had studied this in the lower classes, but am not sure whether this is true or not.

In the figure CD is a median. Does CD divide the angles 1 and 2 in the same ratio of the sides a and b?
=> Is m(1)/m(2) = a/b

Answer 1

No, not true at all.

For example, let AD = DB = 1, and let CD = 0.1. By the triangle inequality, we must have 0.9 < a < 1.1 and 0.9 < b < 1.1, which leads to 9/11 < a/b < 11/9. The ratios of the angles can be any positive number. Clearly, they can't always be the same.

Answer 2

For an easily computed counterexample, take $\angle A$ to be a right angle, take $|AB|=2$, and $b=1$. Then $\angle 2 = 45^\circ$ and $\angle 1 = 135^\circ$, so $\angle 1/\angle 2 = 3$, but $a/b = \sqrt5$.

As mentioned in a comment, you might be thinking of the angle bisector theorem.