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You are trying to solve the ellipsoidal triangle where φ1, α2, s12 are given. (See Figure 1 of the Wikipedia article, Geodesics on an ellipsoid, for the notation.) In your case, we have α2 = ±½π because of the requirement of tangency. This is Problem 7 in §10 of my paper Geodesics on an ellipsoid of revolution (Feb. ...

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