Circle Through Three Points
Center and radius of the circle through three measured points on an arc (the circumcircle - the inverse of bolt-circle): center = circumcenter of the triangle, radius = distance from the center to any point, with the diameter and circumference. Recovers a curve's radius from three field points when the chord and midspan rise cannot be measured directly (first-principles coordinate geometry).
Formula and source
Circumcircle of the triangle on the three points: D = 2(x1(y2-y3) + x2(y3-y1) + x3(y1-y2)); center = ((|P1|^2(y2-y3)+|P2|^2(y3-y1)+|P3|^2(y1-y2))/D, (|P1|^2(x3-x2)+|P2|^2(x1-x3)+|P3|^2(x2-x1))/D); radius = distance(center, P1).
Circle through three points (the triangle's circumcircle) - first-principles coordinate geometry as in Machinery's Handbook (Industrial Press), by name; public domain.
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