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I have a quite tricky geometry problem to solve, and I don't find the answer. Here is the problem:

I have two concentric circles of known radii (r for the inner circle and R for the outer one). Between those circles, there is a segment which cuts the circles. We know the angle (alpha) between the segment and a line perpendicular to the cirlces passing through the point of the segment on the outer circle.

With this, we want to know the angle made by segment and the perpendicular line of the inner circle.

The motivation is to investigate on the refraction of light through circular tubes.

The picture of my problem is given below

Thank you all :)

Simon

Image

  • I didn't mention it but the lenght of the segment D is not known. –  Dec 14 '17 at 14:36
  • If you extend the line $BA$ past $A$ it intersects the smaller circle at a point $A’.$ so you need to specify that angle $BAD$ is not a Ute, if that is your intention. – Aaron Meyerowitz Dec 14 '17 at 15:59
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    Most Anglos are not Native Americans, much less being members of the Ute or Pai-ute tribes. I think Aaron means to say 'acute' instead. Gerhard "Blast That Nosy Interfering Spellcheck" Oarsman , 2017.12.14. –  Dec 14 '17 at 16:57
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    I would use analytic geometry for this problem, letting B be the point with coordinates (R,0), forming the line y=m(x - R) with m the negative arctangent of the known angle, and find the intersection with the equation of points on the small circle. There will often be two solutions for A. When you have the right one, you can use arctangent again on A to help find the angle. This problem is better on math.stackexchange. Gerhard "We Roll Geometry Differentially Here" Paseman, 2017.12.14. –  Dec 14 '17 at 17:06
  • Isn't this a simple (SideSideAngle) triangle trigonometry problem for triangle $AOB$? What am I missing/ – Somos Dec 14 '17 at 21:08

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