Intereting Posts

What good are free groups?
Ring of rational-coefficient power series defining entire functions
Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely
$l^p$ space not having inner product
Pythagorean triples $(a,b,c)$ and divisibility of $a$ or $b$ by $3$.
Fraction field of the formal power series ring in finitely many variables
Probability distribution function that does not have a density function
Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?
When does the product of two polynomials = $x^{k}$?
How do I calculate the probability distribution of the percentage of a binary random variable?
Understanding the Analytic Continuation of the Gamma Function
Number of Multiples of $10^{44}$ that divide $ 10^{55}$
Find ratio / division between two numbers
Visualizing tuples $(a,b,x,y)$ of the extended Euclidean algorithm in a four-dimensional tesseract. Are there hidden symmetries?
How do I find roots of a single-variate polynomials whose integers coefficients are symmetric wrt their respective powers

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.

- Center of circle given 4 points
- A circle with infinite radius is a line
- Minimum operations to find tangent to circle
- For any point outside of a circle, is there ever only one tangent to the circle that passes through the point?
- Is there a way to represent the interior of a circle with a curve?
- How to equally divide a circle with parallel lines?

*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.

- circular reasoning in proving $\frac{\sin x}x\to1,x\to0$
- How many planar arrangements of $n$ circles?
- Did Euclid prove that $\pi$ is constant?
- Determine where a point lies in relation to a circle, is my answer right?
- Point on circumference a given distance from another point
- Area of the intersection of four circles of equal radius
- Calculating circle radius from two points on circumference (for game movement)
- Two Circles and Tangents from Their Centers Problem
- Calculate $\pi$ precisely using integrals?
- Find the differential equation of all circles of radius a

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 $D$.

- Fourier Transform – Laplace Equation on infinite strip – weird solution involving series
- Solve $x^3=y^2-7$?
- How does one prove that $\frac{1}{2}\cdot\frac{3}{4}\cdots \frac{2n-1}{2n}\leq \frac{1}{\sqrt{3n+1}}?$
- Polar to cartesian form of r=cos(2θ)
- generalization of midpoint-convex
- If $E \in \sigma(\mathcal{C})$ then there exists a countable subset $\mathcal{C}_0 \subseteq \mathcal{C}$ with $E \in \sigma(\mathcal{C}_0)$
- Understanding the Gamma Function
- Prove that if $2^n – 1$ is prime, then $n$ is prime for $n$ being a natural number
- Integral of $\sin (x^3)dx$
- Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$
- $PA^2\sin A+PB^2\sin B+PC^2\sin C$ is minimum if P is the incenter.
- Equivalence of the definition of Adjoint Functors via Universal Morphisms and Unit-Counit
- Bayesian Parameter Estimation – Parameters and Data Jointly Continuous?
- What is the interior of a singleton?
- Not a perfect square of the form for any integer x.