Given a triangle's circumcenter, incenter, and foot of one inner bisector, construct its vertices 
Wernick's list problem number 82: The goal is to construct, with a straightedge and compass only, vertexes of $\triangle ABC$ but we're only given:

*

*its circumcenter, $O$


*its incenter, $I$


*its foot of the inner bisector of vertex $A$ on side $BC$, $T_a$.

I draw line $IT_a$ for this is the inner bisector. And that's kinda it. These points are very unrelated. I thought about finding the contact point of the incircle with side $BC$ but this point also has no relation with $O$.
$O$ and $H$ are isogonal conjugates but this doesn't help much
 A: It seems that although this problem is solvable (i.e. constructable), there is no simple construction.
The paper Wernick’s List: A Final Update, which is a survey whether the problems in Wernick's list are solvable or not explains:

[I]t is theoretically possible to extract a construction [$\dots$], but it
is very difficult to obtain and [$\dots$] is geometrically unappealing

In Example 2, this is explained for Problem 108:

Recall that it is possible to perform additions, multiplications,
divisions and root extract by using ruler and compass constructions.
This construction might not be elegant, but it is perfectly valid

So in theory, it is possible to construct Problem 82, but this would involve using a ruler and compass for arithmetic operations and square radicals, which seems out of the spirit of synthetic constructions.
They don't rule out the possibility of an elegant construction, but based on what they say I'd advise you to not spend a lot of time looking for one because even automated geometry solvers have come up empty handed so far.
There is a compendium of automatically generated constructions for Wernick's list at http://poincare.matf.bg.ac.rs/~vesnap/animations/compendiums.html. The construction for $O, T_a, I$ is notably non-existent.
