Is the figure the circumference of a unit circle?

Solution 1:

I've just got the following figure.

The answer for Question is No.

$S$ has eight line segments $AB, CD, EF, GH, MN, OP, QR, ST$ and four points $I, J, K, L$ without twelve points $A, B, C, D, E, F, G, H, M, P, R, T$ where $$A(0,1), B(-\frac12, \frac12), C(-1,0), D(-\frac12, -\frac12), E(0,-1), F(\frac12, -\frac12), G(1,0), H(\frac12, \frac12), I(0,\frac12), J(-\frac12, 0), K(0,-\frac12), L(\frac12,0), M(0,\sqrt{10}), N(\frac{\sqrt{10}}{2},\frac{\sqrt{10}}{2}), O(-\frac{\sqrt{10}}{2},\frac{\sqrt{10}}{2}), P(-\sqrt{10},0), Q(-\frac{\sqrt{10}}{2},-\frac{\sqrt{10}}{2}), R(0,-\sqrt{10}), S(\frac{\sqrt{10}}{2},-\frac{\sqrt{10}}{2}), T(\sqrt{10},0).$$

I think this figure is one of the counterexamples.