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I have a circle with radius $r$ and two line segments extending radii of the circle with length $r+h_1$ and $r+h_2$ respectively (you can think of them being at height $h_i$ over the circle). Forming a triangle from these line segments, the new side is tangent to the circle. I would like to find the length of this new segment (or equivalently, the angle the triangle makes at the center of the circle).

circle with two radii extended by different heights, with the extreme points of the segments joined to make a triangle.

I generated the diagram with GeoGebra (plus GIMP), I hope that makes it easier to understand.

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Drawing a line from the center point to the tangent line of your segment, which we will call $x$. Note that the line from the center to $x$ is perpendicular to the tangential segment, so we can solve for the lengths of the segment on either side of $x$.

enter image description here

On the bottom side, we have $\sqrt{(r+h_2)^2 -r^2} =\sqrt{2rh_2 +h_2^2}$, and on the top side we similarly have the side length $\sqrt{2rh_1 + h_1^2}$. Therefore the length of that segment is exactly $\sqrt{2rh_1 +h_1^2}+\sqrt{2rh_2 + h_2^2}$.

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