Assume the length of $|AB|$ and the following angles are known: $\measuredangle ABD$,$\measuredangle DBC$,$\measuredangle DAC$. Assume it is also known that $\measuredangle ADC=\measuredangle DCA$ (not in the attached figure).
What is the length of $|AD|$? (I guess there may be multiple, countable solutions).
Apply the cosine rule to triangles $DAC$, $DBC$, $ABD$ and $ABC$: you'll get four equations in the four unknowns $AC=AD$, $BD$, $BC$, $CD$.
Eliminating $AC$ and $CD$ is easy: one is left with two quadratic equations in $BC$ and $BD$, having in general four solutions (possibly complex).
I used Mathematica to get an explicit solution but it's too long to be of any use. It's probably wiser to get a numerical solution through some routine.