How to rescale the circle by a factor of CB/XB to make it reach C

Question

Circles tangent to a given line (AB) at a given point (B) are all related by a similarity transformation centered at the point.

Point is given like on this picture.

The problem is... How to reach this idea:

To make one of those circles go through a second point (C in this question), draw any tangent circle at B and its intersection X with line BC, then rescale that circle by a factor of CB/XB to make it reach C.

My question is related to this question.

Answer 1

Let PB⊥AB. The perpendicular bisector of BC will cut PB at K. The required circle is centered at K and radius = KB.

If this is what you are asking for, then the scale factor trick becomes not necessary.