Construction of a triangle, given: side, sum of the other sides and angle between them. Given: $\overline{AB}$, $\overline{AC}+\overline{BC}$ and $\angle C$. Construct the triangle $\triangle ABC$ using rule and compass.

Let $a = BC$, $b = CA$, $c = AB$, $\gamma = \angle C$. We are given $\gamma$, $c$, $a + b$. 
Construct an angle of measure $\gamma$ at a vertex $C$. The problem is to place $A$ and $B$ on the sides of the angle. 
Construct the semi-perimeter $s = \frac{1}{2}[(a+b) + c]$. The points of contact $D_2$ and $E_2$ of the excircle of $ABC$ relative to vertex $C$ with sides $CA$ and $CB$ are at a distance $CD_2 = CE_2 = s$ from $C$. Place these points on the side of the angle and construct the excircle.
Similarly, the points of contact $D_1$ and $E_1$ of the incircle of $ABC$ with sides $CA$ and $CB$ are at a distance $CD_1 = CE_1 = s-c$ from $C$. Place these points on the side of the angle and construct the incircle.
Now the problem is reduced to that of constructing a common internal tangent to two circles, whose solution is well known.

