Related papers: A "right" path to cyclic polygons
Given any positive integer $n$, it is well-known that there always exists a triangle with rational sides $a,b$ and $c$ such that the area of the triangle is $n$. For a given prime $p \not \equiv 1$ modulo $8$ such that $p^{2}+1=2q$ for a…
If the four triangular facets of a tetrahedron can be partitioned into pairs having the same area, then the triangles in each pair must be congruent to one another. A Heron-style formula is then derived for the volume of a tetrahedron…
In the beginning of this paper, we present the general solution to the trigonometric equation asinx+bcosx=c. After that, we focus on the case when a^2+b^2=c^2. In this case, the general solution is expressed in terms of the acute angle…
In his works [R1,R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of…
In this paper we obtain cyclic pentagons and hexagons with rational sides, diagonals and area all of which are expressed in terms of rational functions of several arbitrary rational parameters. On suitable scaling, we obtain cyclic…
We show that the recent derivation that triangleland's topology and geometry is $S^2$ from Heron's formula does not extend to quadrilaterals by considering Brahmagupta, Bretschneider and Coolidge's area formulae. That $N$-a-gonland is more…
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…
It is easy to find a right-angled triangle with integer sides whose area is 6. There is no such triangle with area 5, but there is one with rational sides (a `\emph{Pythagorean triangle}'). For historical reasons, integers such as 6 or 5…
Any permutation-invariant function of data points $\vec{r}_i$ can be written in the form $\rho(\sum_i\phi(\vec{r}_i))$ for suitable functions $\rho$ and $\phi$. This form - known in the machine-learning literature as Deep Sets - also…
This paper investigated the problem of embedding a simple Hamiltonian Cycle with n vertices on n points inside a simple polygon. This problem seeks to embed a straight-line cycle (without bends), which does not intersect either itself or…
This paper introduces an extension of Heron's formula to approximate area of cyclic n-gons where the error never exceeds $\frac{\pi}{e}-1$
This study investigates a generalisation of the Pythagorean theorem to the lengths of conic arcs constructed symmetrically on the sides of a right triangle. It is demonstrated that the theorem remains valid whenever the conic eccentricity…
Visibility graph of a simple polygon is a graph with the same vertex set in which there is an edge between a pair of vertices if and only if the segment through them lies completely inside the polygon. Each pair of adjacent vertices on the…
Recollect that Heron's formula for the area of a triangle given its sides has a counterpart given the medians instead, which carries an extra factor of $\frac{4}{3}$. On the one hand, we formulate the pair of these in Linear Algebra terms,…
An ill-posed problem of synthesis of the Pierce electrodes for a cylindrical beam with a polygonal cross-section is considered. It is assumed that a beam of charged particles is extracted from a space-charge-limited planar diode and the…
We study closed smooth convex plane curves $\Gamma$ enjoying the following property: a pair of points $x,y$ can traverse $\Gamma$ so that the distances between $x$ and $y$ along the curve and in the ambient plane do not change; such curves…
This article proves a Pythagoras-type formula for the sides and diagonals of a polygon inscribed in a semicircle having one of the sides of the polygon as diameter.
A Heron triangle is a triangle whose side lengths and area are all positive integers. If the greatest common divisor of the three side lengths is $1$, it is called a primitive Heron triangle. In this paper, we give an equivalent condition…
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
We provide a new proof of the elementary geometric theorem on the existence and uniqueness of cyclic polygons with prescribed side lengths. The proof is based on a variational principle involving the central angles of the polygon as…