I drew a line through the center points of the circles and perpendiculars to that line through the centers. I then marked the points where the perpendicular lines crossed the circles. These are A,B,C and D.

I drew line AD and marked the point where it crossed the line connecting the circle centers. This is I. This point would be on both of the inner tangent lines.

I then constructed a segment from I to the center of the larger circle and found its midpoint J. I hid line AD because it was no longer needed.

I constructed a circle with center J and segment JI as a radius. I marked the points where this circle intersected the larger circle. These are K and L. These points are also on the tangent lines.

I then drew lines IK and IL, which are the tangent lines I was looking for.

I then hid all of the other construction lines and I was done.