I am struggling to show that constructed triangle is indeed the desired one
Condition: In triangle $ABC$, one has marked the in-center, the foot of altitude from vertex $C$ and the center of the ex-circle tangent to side $AB$. After this, the triangle was erased. Restore it.
Analysis: Let $I$ be the in-center and $I_C$ be the ex-center and $H$ be the foot of altitude from $C$.
There are several ways of showing that $AB$ is the angle bisector of $\angle IHI_C$. This would restore the side $AB$. Lines $II_A$ and perpendicular to $H$ on $AB$ would intersect at point $C$.