Draw a circle with intersection, tanget & radius

So I'm having trouble trying to create a Circle with a fixed Radius that is tangent to a circle and crosses an intersection. See image below. I can't for the life of me figure out how to do this. Any help is greatly appreciated! IMG and DWG attached.

geometry.dwg

A tangent is always 90 from a line drawn to the centre of a circle, if I understand correct draw a line from intersect point back to center, draw a circle of correct radius at this point, using intersect line to new circle just move the new circle from this point to the correct point.

Or you could offset the smaller circle by the radius of the larger circle. Draw a circle, same radius as the larger circle and centred at the intersection point. Your larger circle should be centred at the intersection of these two.

Or you could offset the smaller circle by the radius of the larger circle. Draw a circle, same radius as the larger circle and centred at the intersection point. Your larger circle should be centred at the intersection of these two.

This works and I don't know why, and it's killing me

You can set the UCS for the middle of the circle with the X axis aimed to the intersection. Move the circle with the quadrant osnap, go back to WUCS. No extra lines or circles required.

You could also move the small circle to the large one so their quadrants are touching, then use rotate with reference option. No extra lines or circles required.

To draw a circle tangential to another circle, you know that the centre of that circle must be at a distance equal to the radius from the first circle. So if you offset the small circle by the radius of the large circle, the centre of the large circle must lie somewhere along the offset line.

To draw a circle that passes through a point, if you draw a circle centred on that point with a radius equal to that circle, then the centre of the circle must lie somewhere along that circle.

If you combine the two, the intersection of the two circles gives a point that makes the circle tangential to the first circle and passes through the required point.