One circle, two lines Apollonius' problem

Question

I've been trying to solve special case of Apollonius' problem, where instead of 3 circles i have 1 circle and 2 lines. Acording to:

http://en.wikipedia.org/wiki/Special_cases_of_Apollonius%27_problem#Type_8:_One_circle.2C_two_lines

First I should shrink the circle to a single point. I've done it and that single point is the centre of the original circle. Now things are easier, because I need to solve Apollonius' provlem for 2 lines and a point, which is fairly simple. But when I'm finished with this I encounter a problem with shrinking or expaning the solution circle.

I've tried using homothetic transformation and using the intersection of the both lines as homothetic center. And now the line that connects the homothetic center and the center of the circle intersect the circle and that should be the point where the circles touch each other, but instead a very small area is shared by both circles, after I'm finished with the homothetic transformation.

I think that this isn't problably the right way to modify the solution circle, because in order to touch, not to intersect the original circle, the touching point should be on the line that connects both circles' centers. But I don't know how to find that point because, the center of solution circle is not fixed(it's moving while i'm shrinking or expanding the the solution circle).

Answer 1

When you shrink your circle to a point, you should also move your lines by the radius of that circle. Then the reverse operation, exapnding the point to a circle and moving the lines back, will not change the center of the solution circle, but only its radius.