Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2510.15918

The One-Seventh Area Triangle in the Complex Plane: Proving the Proportionality Using Features of the Complex Number System

Mathew Miltonhardy
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper explores and proves the one-seventh area triangle using a purely algebraic approach as opposed to a geometric one. A triangle set purely in the complex plane is used so that we can utilise features of the complex number system to determine areas. The area of a triangle in determinant form is derived using these features, and a rigorous proof of the one-seventh area triangle follows.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mathew Miltonhardy. 2025-09-30. The One-Seventh Area Triangle in the Complex Plane: Proving the Proportionality Using Features of the Complex Number System. https://arxiv.org/abs/2510.15918

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The new Fermat-type factorization algorithm

Let n be any odd natural number other than a perfect square. We show that the new factorization algorithm, presented in this paper and which we call DFM-1 (where DFM stands for Detto's Factorization Method), is much more efficient than the implementation technique of Fermat's Factorization Algorithm (FFA) called FFA-1, which, among the implementation techniques of Fermat's Factorization Algorithm (FFA), is the one that requires the fewest iterations to identify the non-trivial and trivial factors of n (excluding the cases in which the two factors of the pair of non-trivial or trivial factors of n are so close to each other that they can be identified at the 1st iteration with each of such implementation techniques). Indeed, by the way in which Euler's totient function of any n that is a semiprime is applied to FFA-1, we arrive at the new factorization algorithm (DFM-1), which halves (possibly rounding up to the next integer) the number of iterations required by FFA-1. Furthermore, in this paper, we present the hypothetical scenario according to which the number of iterations could possibly be further reduced. Finally, and still in relation to this new factorization algorithm, in this paper we present the limit number of iterations, which is less than the number of iterations required by DFM-1 to reach the condition x - y = 1 which characterizes the pair of trivial factors of n, beyond which it is no longer possible for pairs of non-trivial factors of n to occur.

math.GM↗
arXiv2026

A Note on the Measure of Vector and Pythagorean Theorem

Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, $\textit{canonically}$ stems from a $\textit{single}$ definitional primitive $-$ the ($\mathbb R^{\vcenter{\hbox{$\scriptscriptstyle+$}}}\!$-quantitative) invariant $\mathcal Q$-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared $p = |\mathfrak a|^2$ $-$ the quantum Born rule $-$ and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the $\textit{well-defined}$ interpretable turns out to be, up to gauge $\mathcal Q {\,\to\,} \mathrm{const} {\,\vcenter{\hbox{$\scriptstyle\times$}}\,} \mathcal Q$, the unique gauge-invariant measure $\mathcal Q=|\hspace{-0.18em}| {\cdot}{\cdot}{\cdot} |\hspace{-0.18em}|^2$; independently of the field $\mathbb R$ or $\mathbb C$.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

math.GM↗