Compass only non-collapsing compass (Euclidea Series)

Since I've branched into this topic, I want to include some notes on additional references I've found and some results I've been able to get. First, an admission:

I'm still having trouble finding the intersection points of a circle and a line when the circle's center is on the line!

Another resource
James King (UW home page) has a nice session outline working through compass only constructions. Be warned, there are some spoilers for some of the Euclidea challenges in that material. Unfortunately, I can't tell from write-up when and where this was used. Maybe this NWMI meeting?

In Professor King's notes, he describes the construction I'm struggling with as "important and difficult," so I will have to redouble my efforts.

Non-collapsing compass from collapsing compass
As an aside, perhaps you have noticed that everyone else refers to the two types of compasses as collapsible and non-collapsible. Doesn't that terminology strike you as wrong, too? The point isn't that one type of compass is able to collapse, but rather that it always collapses when not drawing a given circle. Also, a common form of "non-collapsible" compass is able to collapse!

Just squeeze the legs together to collapse

Anyway, I was able to figure out the construction of a non-collapsing compass, just in time for Euclidea 13.2.

I'll give a construction below. For a more mild hint, the key is the ability to reflect points through a line we've already "constructed" (where we have two points on the line).

I've chosen to do this in GeoGebra so I can show a sequence, explain some of the reasoning, and grey out earlier parts of the construction that we don't need anymore. Check my work to make sure I didn't slip in any subtle straightedge moves!

Here, we are given points A, B, and C and want to construct the circle centered at B with radius AC.

First step is two circles centered at A and B with radii AB, giving us points D and E. The line DE will be the one we are reflecting through. Key is that A and B are the reflected images of each other through this line.

Next, we draw the circle at A with radius AC. This intersects our circle centered at B in point F. F is a special point, distance AC from A and AB from B, and distance DF from D.

That means the reflection of F through line DE is a point distance AC from B! We can find it using the fact that it will also be distance AB from A and distance DF from D:

It almost looks like C and F are the same distance from D. That isn't true and is entirely coincidental. Try it out yourself and use the move tool to push C around and see that this is the case. In particular, you might notice that we could have made another choice for F (it is a point of intersection of two circles, we could have taken the other) and that other choice is on the opposite side of AB from C.

I'll leave to you the satisfaction of completing the construction and drawing our target circle.

Also, the Euclidea challenge is formulated very slightly differently: instead of point C, we are given the starting circle centered at A. That means the construction is one move shorter than what I've shown here.

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