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[4] viXra:2106.0174 [pdf] submitted on 2021-06-29 23:18:35
Authors: Hiroshi Okumura
Comments: 3 Pages.
We consider Enomoto's problem involving a chain of circles touching two parallel lines and three circles
with collinear centers. Generalizing the problem, we unexpectedly get a generalization of a property of the power of a point with respect to a circle.
Category: Geometry
[3] viXra:2106.0173 [pdf] submitted on 2021-06-30 00:10:47
Authors: Hiroshi Okumura
Comments: 34 Pages.
We demonstrate several results in plane geometry derived from division by zero and division by zero calculus. The results show that the two new concepts open an entirely new world of mathematics.
Category: Geometry
[2] viXra:2106.0165 [pdf] replaced on 2024-03-22 18:50:45
Authors: Theophilus Agama
Comments: 10 Pages. This paper has been technically and substantially improved.
In this paper we show that the number of points that can be placed in the grid $ntimes ntimes cdots times n~(d~times)=n^d$ for all $din mathbb{N}$ with $dgeq 2$ such that no three points are collinear satisfies the lower boundbegin{align}gg n^{d-1}sqrt[2d]{d}.onumberend{align}This pretty much extends the result of the no-three-in-line problem to all dimension $dgeq 3$.
Category: Geometry
[1] viXra:2106.0158 [pdf] replaced on 2021-09-10 08:10:35
Authors: Theophilus Agama
Comments: 8 Pages. Minor tweak in the lower bound
Let $\mathcal{R}\subset \mathbb{R}^n$ be an infinite set of collinear points and $\mathcal{S}\subset \mathcal{R}$ be an arbitrary and finite set with $\mathcal{S}\subset \mathbb{N}^n$. Then the number of points with mutual integer distances on the shortest line containing points in $\mathcal{S}$ satisfies the lower bound
\begin{align}
\gg_n \sqrt{n}|\mathcal{S}\bigcap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]|\sum \limits_{\substack{k\leq \mathrm{max}_{\vec{x}\in \mathcal{S}\cap \mathcal{B}_{\frac{1}{2}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]}[\vec{x}]}\mathcal{G}\circ \mathbb{V}_1[\vec{x}]\\k\in \mathbb{N}\\k>1}}\frac{1}{k},\nonumber
\end{align}where $\mathcal{G}\circ \mathbb{V}_1[\vec{x}]$ is the compression gap of the compression induced on $\vec{x}$. This proves that there are infinitely many collinear points with mutual integer distances on any line in $\mathbb{R}^n$ and generalizes the well-known Erd\H{o}s-Anning Theorem in the plane $\mathbb{R}^2$.
Category: Geometry
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