Can we assume that is perpendicular to at the point of tangency D?

Draw OX and OY perpendicular to AC and AB, respectively, at their points of tangency.

Center O is the incenter of the inscribed circle found by the intersection of the three angle bisectors. Thus,

Angle ACO = Angle BCO

Angle CAO = Angle BAO

Angle ABO = Angle CBO

CD = CX and BD = BY, since tangents to a circle from an external point are equal.

CX = 6

BY = 8

Use Arctan to find angles OCD and OBD.

Angle OCD =

Angle OBD =

Since the angles at C and B were bisected,

Angle ACB = 2(33.69) = 67.38

Angle ABC = 2(26.57) = 53.13

This makes Angle CAB = 180 - 67.38 - 53.13 = 59.49

Half that gives Angle CAD = 29.75

Use Tangent to find AX.

AX = AY because they are tangent segments from the same external point.

Therefore, AC = 13 and AB = 15, approximately.