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SUBMIT YOUR IDEAWow - that was somewhat painful.
Started out with a trig approach, and in doing so spent 4 hours trying to debug radians vs. degrees and kilometers (turns out that I was using the same math as @Joe_Mako except Joe got it to work)
Then, like @LordNeilLord and @NicoleJohnson I did a strategic retreat and used the same technique used by others ( @gnans19 was first to post this method I think?)
This has also spawned a few new product requests:
https://community.alteryx.com/t5/Alteryx-Product-Ideas/Useful-Spatial-formulae/idi-p/85206#M4794
I tried (and struggled) using the spatial tools, then wrote loads of calcs in the formula tool.
Solution attached.
I loved this challenge.
In fact, I enjoyed it more than any of the 80 other challenges I've done so far, which is a little weird considering most of the hard work happened in my head while staring at a piece of paper. Once I had an approach figured out, making it work in Alteryx was beautifully quick.
As a non-mathematician, you know something special is happening when your desk looks like this and your scribbled notes end up delivering a perfect result. Magical.
Here's my solution:
Update: Comparing against other solutions, it's interesting to see that I could have used distance to select the optimal point from the split trade areas around the start and end point. I think I probably made it harder for myself by relying on the angle of the line to work out the intercept points (especially given the fact that point 0 in a split circle is at the 90 degree point which makes the math a little counter-intuitive) but even so this feels like a pleasing solution.
I attempted this challenge a while ago but got stuck with trigonometry and left it for later. Here is my solution, definitely overcomplicated one