Here's a solution for circles with different radius that doesn't require right angle measurement or parallel lines:
Draw a tangent on the larger circle
Draw two tangents on the smaller circle that intersect where the first tangent touches the larger circle
Draw two tangents on the larger circle where the tangents from step 2 intersect the larger circle opposite to the first tangent
Find the intersection of the tangents from the 3rd step. A line from this to where the first tangent touches the larger circle must go through the center of the two circles.
Repeat the above with the first tangent intersecting on a different point on the larger circle. The intersection of the lines from the 4th step is the center of the circle.
Here's a solution for circles with different radius that doesn't require right angle measurement or parallel lines:
Edit: visual aid
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