I would be very surprised if someone could get very far by just eyeballing these! The pictures get complicated with even a few circles—I think the only way to understand them is to start with the two given points on a blank piece of paper, and draw the circles one at a time (figuring out which circle can get drawn next based on the points available).
But then, I'm a very analytical sort of person. I get overwhelmed very easily—no less in life generally than in geometric problems!—so I'm pretty used to breaking things down into steps and taking them one-at-a-time.
That program, incidentally, was very tricky to put together! The reason is that every time you draw a circle, you add a LOT of new points where new circles can be drawn... so the number of possibilities grows extremely quickly, meaning it takes a long time to search them all. My first attempt took around 400 seconds to solve Napoleon's problem, which just has six circles, and solving problems with any more than that were out of reach. I think I'm on something like version 20 of that program, now, and it's able to solve Napoleon's problem in 40 milliseconds—some five orders of magnitude faster—but even then it's only really up to tackling problems involving up to maybe twelve circles. Sort of the holy grail I've been aiming at working out is Gauss' masterpiece, the heptadecagon, but my program's not up to the job, and so I'm trying to understand what the constructions mean better before I can make the attempt.
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But then, I'm a very analytical sort of person. I get overwhelmed very easily—no less in life generally than in geometric problems!—so I'm pretty used to breaking things down into steps and taking them one-at-a-time.
That program, incidentally, was very tricky to put together! The reason is that every time you draw a circle, you add a LOT of new points where new circles can be drawn... so the number of possibilities grows extremely quickly, meaning it takes a long time to search them all. My first attempt took around 400 seconds to solve Napoleon's problem, which just has six circles, and solving problems with any more than that were out of reach. I think I'm on something like version 20 of that program, now, and it's able to solve Napoleon's problem in 40 milliseconds—some five orders of magnitude faster—but even then it's only really up to tackling problems involving up to maybe twelve circles. Sort of the holy grail I've been aiming at working out is Gauss' masterpiece, the heptadecagon, but my program's not up to the job, and so I'm trying to understand what the constructions mean better before I can make the attempt.