The least possible radius is obtained when the two given points are the endpoints of the diameter of the circle. Therefore, the smallest radius equals half the distance between the two points.
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The circumcentre is always equally distant from all three vertices of a triangle. Therefore, OA, OB and OC are equal. Small differences may appear only because of minor drawing or measuring errors.
The triangle formed is an obtuse-angled triangle. In every obtuse-angled triangle, the circumcentre lies outside the triangle. Therefore, the centre of the circumcircle is outside the triangle.
The triangle formed is an acute-angled triangle. In every acute-angled triangle, the circumcentre lies inside the triangle. Therefore, the centre of the circumcircle is located inside the triangle.
The total outer side length including the path becomes 40 + 2s metres. Subtracting the inner square area (1600) from the expanded outer square area provides the final algebraic expression for the path.