That nephew must have access to a CAD program with constraint feature :)

x is equivalent to a scale factor. So x = 1 or 10, or even 100 is good.

The base chord, (x*sqrt(2)), can be generated as the hypotenuse of a 45 45 90 triangle with two sides of x*1 unit(s), by pythagoran.

The fact that 4 chords are of equal length x*1 units, implies that their central angles are equal, so the rotation angle from one to the other is equal,
with a value of approximately 114.81 degrees +/-. There are 3 of these equal angles, I am calling phi.

The angle at A2, of approximately 97.78, (I'm calling it theta), must be adjusted so that all vertices fit on the 3point circle.
Every time angle theta is adjusted, the angle phi also changes, by the simple formula previously given. So all of the x*unit chords keep changing as the candidate theta value changes. So they need to be redrawn over and over.
Constraining the 3 angles phi to a function of theta, would solve the puzzle, if the Cad program had constraints, and correct snap(s).

I guess a MoI script or a node could be written with an epsilon of the error of intersecting the centerline at C with chordBC?
Keep running the adjustment of theta, until epsilon reaches zero?

- Brian

My 2018 Alibre program is in a zip file.

I was able to run Barry's node program. (After replacing Display node. Either my copy or Barry's must be old?)
It draw's close approximations to the solution, subject to visual adjustment to approach a solution. (As far as I can tell.)

I do not have enough knowledge of MoI to go much further.

Trigonometry can replace phi with a function of theta, but the resulting equation would still likely require numerical methods to solve. (?)