Related papers: Area Inside A Circle: Intuitive and Rigorous Proof…
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…
This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non-overlapping circles. The first half of the article is an exposition of…
The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…
This article is concerned with the representation of curves by means of integral invariants. In contrast to the classical differential invariants they have the advantage of being less sensitive with respect to noise. The integral invariant…
This communication contributes to research on proof validation as a lens for uncovering didactical phenomena related to proof and proving. We revisit the puzzling case of lower secondary students in France who validate circular proofs. That…
In this paper we consider the problem of packing a fixed number of identical circles inside the unit circle container, where the packing is complicated by the presence of fixed size circular prohibited areas. Here the objective is to…
We reconsider Archimedes' evaluations of several square roots in 'Measurement of a Circle'. We show that several methods proposed over the last century or so for his evaluations fail one or more criteria of plausibility. We also provide…
The Six Circles Theorem of C. Evelyn, G. Money-Coutts, and J. Tyrrell concerns chains of circles inscribed into a triangle: the first circle is inscribed in the first angle, the second circle is inscribed in the second angle and tangent to…
John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present…
The paper reports a generalized expression for filling the congruent circles (of radius r) in a circle (of radius R). First, a generalized expression for the biggest circle (r) inscribed in the nth part of the bigger circle (R) was…
We formalize Pick's theorem for finding the area of a simple polygon whose vertices are integral lattice points. We are inspired by John Harrison's formalization of Pick's theorem in HOL Light, but tailor our proof approach to avoid a…
We illustrate the concept of mathematical proof.
We prove that a surface in real 3-space containing a line and a circle through each point is a quadric. We also give some particular results on the classification of surfaces containing several circles through each point.
Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles…
We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane.
In this paper we develop cyclic proof systems for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions belong to the quantifier-free fragments…
This paper frames calculus as a global, centuries-long development rather than a subject that began only with Newton and Leibniz. Drawing on ideas from Greek, Indian, Islamic, and later European mathematics, it highlights how concepts like…
Heron's formula states that the area $K$ of a triangle with sides $a$, $b$, and $c$ is given by $$ K=\sqrt {s(s-a) (s-b) (s-c)} $$ where $s$ is the semiperimeter $(a+b+c)/2$. Brahmagupta, Robbins, Roskies, and Maley generalized this formula…
We give a new proof of a lemma by L. Shepp, that was used in connection to random coverings of a circle.
We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this…