Relation between angles and sides of an Isosceles triangle Please refer to this image:

$A = (0,0)$  $B = (4,0)$ and $C=(2,3)$ forms an isosceles triangle and let $\theta$ be the $\angle C$. Now I know if $D$ point bisects $AB$, then it also bisect the angle $\theta$.
My Question is:
If $AE = \frac{AB}{3}$, does $\angle ACE$ (lets say $\alpha$ as shown) has any relation to the original angle $\theta$?
$\alpha = k \cdot \theta$, then k= ?
Does $k=\frac{1}{3}$  still holds true$?$
If so how? I may be very stupid asking such elementary question. But am stuck in it for long, and might be missing something very silly. Any help is appreciated.
 A: No, the ratio does not hold in general.
You can prove this to yourself very simply if you extend the base further and further to the right, maintaining the original location of the left vertex and the "top" vertex, such that angle ACE retains its original measure.
For instance, if you extend side AB such that the new position of corner B (call it $B'$) makes segment $B'E$ longer than segment AE by a factor of 1000 times, will angle $ECB'$ be 1000 times the measure of angle ACE?  Obviously this is impossible, since no matter what, angle $ECB'$ will be less than 180 degrees.

You can work out relationships (read up on trigonometry), but the rule you cite for isosceles triangles doesn't generalize in the way you are exploring.
A: No it does not. If you work out the angles using elementary trigonometry you find
$$ \alpha = \frac{\theta}{2} - \tan^{-1}\left(\left(1-2x\right)\tan\left(\frac{\theta}{2}\right)\right),$$
where $x = \frac{\overline{AE}}{\overline{AB}}$ is the fraction that is cut from the base line. Clearly, for $x=\frac{1}{2}$ you have $\alpha=\frac{\theta}{2}$, however, for other $x$ the ratio is not linear.
For small $\theta$ it is approximately linear though. Here is a plot to show this dependence for different values of $\theta$:

Solid lines represent the exact expression, dashed lines what we would expect for a linear dependence. As evident from the plot for small $\theta$, they agree well but for larger angles they do not.
edit: Wildcard was faster but I'd like to leave the answer in for the illustrative plot.
