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Dear Andrés,

Dear Peter,

Thanks to your published curves Andrés which I took a close look at, and thanks to your Magic Method Peter, I think that I got a solid reply which should cover all possible cases of our initial challenge.

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To solve for the existence of unique or multiple existing Departure Point(s) with a given Great Circle Departure Track (Azimuth) "Z" and a given Great Circle Distance "D" towards one given Arrival Point with Coordinates Lat A / Lon A , we need to consider (1) : Z and (2) : D. and (3) : [ 90° - Lat A ] . 

When and only when the following 2 conditions are both fulfilled : Z ˂ [ 90° - Lat A ]  and D ˃ [ 90° - Lat A ], there are 2 and only 2 Departure Points such as in this example which was very cleverly solved by Peter.

In all other cases, such as Dave Walden's initial Problem, there is 1 and only 1 such Departure Point.

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You are most welcome to attempt defeating this rule. Just one plain counter-example makes it invalid.

Best Isoazimuthal Regards to all,

⚓  Kermit - antoine.m.couette[at]club-internet.fr ✈