The desired construction would be circle equidistant from AB and AC. That is, its center would lie along the line of the angle bisector. It would be slightly further away from A than the center of the Incircle.

It is relatively easy to construct a circle tangent to three different lines. There are four such circles -- the incircle and the three excircles.

Constructing a circle tangent to three lines (L, L, L) and a circle tangent to two lines and a circle (L, L, C) are two of the subproblems of the Problem of Apollonius. Is there anything you can use from the L, L, C part of that problem?

In general, do you have (or can you divise) a way to contruct a circle tangent to two intrsecting lines and a circle?

HINT:   The geometry of inversion may be useful.  Since inversions maps circles through the center of inversion into lines, an appropriate circle of inversion could map the circumcircle into a line intersecting two sides of the triangle.   Then the L, L, L construction would produce a circle  and that circle could be mapped by a reverse inverstion into the desired circle.   THERE ARE OTHER CONSTRUCTIONS.