Reduce a circle while keeping all points inside

Question

I'm trying to reduce the size of a circle but keeping all the points inside it.

Points A and C are part of the circumference S. Segment Line F connects points A and C. Line R is perpendicular to F and passes through the center of the circle, at point G. I want to push the center G towards the intersection of F and R to reduce the size of S. However I need to keep point D inside the circle so I can't move G all the way towards that intersection.

I want to find out how to calculate how much I can move G in the direction of that intersection while keeping D inside the circle.

Note that when moving G towards the intersection the radius will reduce as the goal of my program is to get the smallest circle that contains all the points.

I am aware this is the Smallest-circle Problem. What Im doing is treating a special case of my algorithm that tries to solve that exact problem.

Any ideas? I've been comparing vectors of how much G can move but haven't figured out anything yet.

Answer 1

Then new point G is on the mediatrix of DA.