Cone tangent to two planar faces


I need to build a cone whose tip is at the coordinates (x, y, z), whose tip aperture is A degrees and whose surface is to be tangent to a couple of surfaces: more specifically, in my case I have two adjacent planes, i. e. two planar TopoDS_Face with an edge in common, and the cone has its tip coincident with one vertex of this edge. Is there a way to build (when possible, of course) such a cone or should I get its axis direction via numerical approximation by, for instance, creating the cone with the tip already in the desiderd vertex, and tilting it until I get some intersection between the conical surface and the adjacent planes?

I hope my description of the problem is clear enough, however I also added a picture to explain better my situation.

Many thanks to everyone in advance.