Do the tangents of two circles define concentric circles?

Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$.

Draw the four tangents between $R_1$ and $R_2$. There will be two tangents that cross between $R_1$ and $R_2$ and two tangents that does not cross between $R_1$ and $R_2$. Call the two tangents that cross inner tangents and the two tangents that do not cross outer tangents.

I assert that there are two concentric circles that can be drawn, $C_1$ and $C_2$. $C_1$ will have the four tangents points of the inner tangents on its circumference and $C_2$ will have the four tangent points of the outer tangents on its circumference.

I remember solving this problem using high school geometry, basic algebra and some trig, but that was over $20$ years ago.

Is my assertion correct? If so, what is the solution?

I vaguely remember that one key point was noting that radii that intersect at tangent points are perpendicular to the tangent line.

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