So I have begun making many mistakes, so time to call it for the night.

The circle is approached by successive approximation.
The Radius of the circle is slightly larger that 0.928 * X. (And less that 0.9322 * X.)
The Angle of the base chord of sqrt(2), to the right lower leg, is slightly larger than 97.758235 degrees.

The unit chord to unit chord angle is (slightly larger I think), than 114.819736 degrees.

Place base chord of sqrt(2) units. Its end points are two points on many 3 point circles.
Place start of right leg chord segment of 1 unit, (with perpendicular bisector), at right end of base chord, at an angle of 97.758235 degrees. Its upper end point is the 3rd point on the candidate circle. Create said circle.

Using Orient LineLine, with copy checked, move a copy of right leg with perp bisector, with lower endpoint and upper end point, to upper end point, towards the upper center of the candidate circle.
The 1 unit is SLIGHTLY too long. Left hand two chords can be mirrored, but to not add anything to the proceedure.
So the candidate circle is slightly too long.

Rotate right leg chord segment and perp bisector, to slightly larger angle than 97.758235, and repeat proceedure.


https://imgbox.com/T04618V9

- Brian

I thought History might be helpful, (or not).