TITLE: Given a triangle find the length of BC
QUESTION [7 upvotes]: Given an ABC triangle , if $AB+AD=4$, find the length of BC.

REPLY [6 votes]: We are given that $∠BCD=θ$ and $∠BAD=2θ$. Also, $BD$ is the angular bisector of $∠ABC$.
We draw the circle $\Gamma$ with center $A$ and radius $|AD|$. $G$ is the intersection of $\Gamma$ and $AB$ such that $A$ is between $G$ and $B$. We know that $\angle AGD=\angle ADG$, because $|AD|=|AG|$, thus $\angle GAD=180^\circ-2\angle AGD$.
\begin{align}
\angle BCD&=\frac 12\angle BAD\\
&=\frac 12(180^\circ-\angle GAD)\\
&=\frac12(180^\circ-(180^\circ-2\angle AGD))\\
&=\angle AGD
\end{align}
Thus, $\angle BCD=\angle AGD$, thus
$$|BC|=|BG|=|BA|+|AG|=|BA|+|AD|=4$$