Optical Comparator

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Can u solve this problem?

http://www.box.net/shared/9izlv5oy4c

I need to know the length of line AB. Say we know the length of lines BC, BD and the diameter of the circle.

The reason why I need to solve this is because I work at a quality control place and we have an optical comparator where we can see the outline of an object and measure distances on that outline, and sometimes we need to check how concentric two cylinders are, so I want to be able to check this by having the distance from the edge of one cylinder to the edge of the other from two different angles. So in the graph, the lines BC and BD are the distances from the edge of one cylinder (the circle) to the center of the other one (point B) from two different angles.

I was able to solve this by rotating the object exactly 90 degrees, but a formula that works by rotating it any angle would be more accurate.

If the tangents to the circle are parallel pair-wise, AND if AB is parallel to CD, AND the angle, α, between the parallel lines is known,
AB^2 = BC^2 + BD^2 - 2(BC)(BD)cosα

If AB is parallel to the tangent line through C, then
AB = BCtanα

If α is unknown, you need 1 additional measurement for a solution.

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