>>9243431Not possible in the plane, but in the 3-dimensional space you can have arbitrarily many points with these conditions.
Pick n points on a line (say, the x-axis).
Pick n choose 2 dfferent planes that intersect in the line.
Pick a length that is suitable to connect every two points by an arc of said length - if the length is given, you need to scale/choose the points according to this length.
On each plane, draw an arc / a curve of the given length that connects two of the points on the line and does not contain any other point of the line - repeat this for each of the n choose 2 pairs of points.
Since the planes only intersect in the line, two arcs can only intersect in one of the picked points.
