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A friend of mine taught me the following question. I’ve never heard such a strange and interesting question!

**Qustion**: Supposing that a figure $S$, which is constituted by points, satisfies the following four conditions, can we say that $S$ is the circumference of a unit circle?

*1.* $S$ crosses at **two** points every line which passes through the origin.

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*2.* $S$ crosses at **one** point every tangent of the unit circle whose center is the origin.

*3.* $S$ crosses at **two** points every line $L_x$ which satisfies the following two:

(i) $L_x$ is parallel to $x$-axis. (2) The distance $d_x$ between $L_x$ and $x$-axis satisfies $d_x\lt1$.

*4.* $S$ crosses at **two** points every line $L_y$ which satisfies the following two:

(i) $L_y$ is parallel to $y$-axis. (2) The distance $d_y$ between $L_y$ and $y$-axis satisfies $d_y\lt1$.

I’ve tried to solve this, but I’m facing difficulty. He said the answer is **NO** without his memory of the very figure. Can we find a special counterexample?

*Edit* : I’m sorry. The **Question 2** is not an appropriate question, so I deleted it.

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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.

**EDIT**: This figure is not quite correct as a counterexample.

I managed to construct this counterexample. It is very, *very* “hackish” and contrived. It is probably not what your friend had in mind. But it *is* a counterexample!

Anything in $\color{blue}{\text{blue}}$ is not part of the actual figure.

- Define (only this is in polar coordinates) $R(\varphi):=(\sqrt{2}, \varphi)$. Then the arc in first quadrant is defined as the set of points $\{R(\varphi):0 < \varphi < \pi/2\}$. Whereas the arc in third quadrant is defined as the set of points $\{R(\varphi):\pi < \varphi < 3\pi/2\}$.
- $A = (0, A_y)$ such that $A_y < 1$ is infinitesimally close to $1$.
- $B = (B_x,0)$ such that $B_x > -1$ is infinitesimally close to $-1$.
- $C = (C_x, 0)$ such that $C_x < 1$ is infinitesimally close to $1$.
- $D = (0, D_x)$ such that $D_x > -1$ is infinitesimally close to $-1$.
- $P_1 = (-\sqrt{2}/2, \sqrt{2}/2)$.
- $P_2 = (\sqrt{2}/2, -\sqrt{2}/2)$.

Please critique this figure.

Some comments:

- You might be worried that the lines $y = 0$ and $x = 0$ would violate property 1, 3 and 4. They do not, because the arcs do not actually touch the axes.
- Would the tangent lines $y = x \pm \sqrt{2}$ violate property 2? No, because, again, the arcs do not touch the axes and those tangent lines touch one and only one of the points $P_1$ and $P_2$.
- Each of the tangent lines $y = \pm 1$ crosses only at one point. They pass through neither point $A$ nor $D$.

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