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.