I saw the image and wondered if I could do it without math, meaning without calculating the length of the radius for example.

I don't know if you'd have to "calculate" but you'd probably have to do something measurable (even if it's just "duplicating" lengths with the compass, etc). Geometry is "metry" which is "measuring". Maybe do something based on the picture here:

http://www.artofproblemsolving.com/wiki ... Problem_21
To be more precise, I'm looking to find the point that would be tangent to the large and small circles, instead of just guessing.

Points are coordinates, so this is even more mathematical ("calculating") than just doing the drawing. You'll need to start by establishing a coordinate system, determining the radii, etc, and then maybe even doing some calculus. But at least you've got some messy algebra and coordinate geometry ahead of you.