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Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear) $x:y:z$. Eight unary operations discussed in this paper include $u_1\left(X\right)=(y-z)/x:(z-x)/y:(x-y)/z$. For each $u_i$, there exist, formally, two points,...

This article points out the kinematics in tracing a Bézier curve defined by control mass points. A mass point is a point with a non-positive weight, a non-negative weight or a vector with a null weight. For any Bézier curve, the speeds...

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that compr...

In the paper, we consider holomorphically projective mappings of n-dimensional pseudo-Riemannian Kähler and hyperbolic Kähler spaces. We refined the fundamental linear equations of the above problems for metrics of differentiability class $C^{}$...

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space ${\mathbb{R}}^n$ with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode t...

This article contains a synthetic proof of the following proposition: consider a conic $c_1$ and its variable chord $AB$, which subtends a right angle at a given point P. Then, the foot E of the perpendicular dropped from P onto the line $AB$ lies on a cert...

In this paper, we consider six homogeneous manifolds $GL(n,\mathbb{R})/O(n,\mathbb{R})$, $SL(n,\mathbb{R})/SO(n,\mathbb{R})$, $Sp(2n,\mathbb{R})/U\left(n\right),$$(GL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/O(n,\mathbb{R})$, $(SL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/SO(n,\mathbb{R}),$$(Sp(2n,\mathbb{R})⋉H_{\mathbb{R}}^{(n,m)})/(U\left(n\right)\times S(m,\mathbb{R})).$ They are homogeneous manifolds which are i...

We continue Riemann’s program of geometrizing physics, extending it to encompass gravitational and electromagnetic fields as well as media, all of which influence the geometry of spacetime. The motion of point-like objects—both massive an...

This paper presents a theoretical framework for constructing generalized weighted Voronoi diagrams (GWVDs) of weighted points and straight-line segments (“sites”) in the Euclidean plane, based on lower envelopes constructed in three-dimen...

The cycloidal family of curves in ${\mathbb{R}}^2$, also known as the trochoid family, are equivalently generated by two classic methods: bungee cords and rolling wheels. What about their counterpart families in the hyperbolic unit disk? We review the two methods...

Let S be the $\mathbb{N}\times \mathbb{N}$-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra $k\left[S\right]$, together with the evolution equations of the deformations of the powers...

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths $a,b,$ and c of a triangle $ABC$. For example, the circumcenter is represented by the polynomial $a(b^{}$...

In this article, we provide elementary proofs of two projective generalizations of Hagge’s theorems. We describe Steiner’s correspondence as a projective generalization of inversion.

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in ${\mathbb{R}}^2,$ but it does not in ${\mathbb{R}}^n,$ for $n>2.$ Lord Kelvin overcame this problem by defining a new (at the time...

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case,...

In the ancient realm of geometry, we have witnessed the ultimate display of mathematical abstract thought [...]

We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to the forme...

We prove that if we start with a non-continuous absolute plane, remove Side-Angle-Side as an axiom, and replace it with the three new axioms, Side-Angle-Angle, angle addition, and the existence of angle bisectors, then the result is also an absolute...

It is well known among geometry scholars that the golden triangle, an isosceles triangle with sides and base in golden ratio, maintains a significant relationship with regular polygons, notably the regular pentagon, pentagram, and decagon. Extensive...

In recent years, two quite different tools have been employed to study global properties of arcs in the plane. The first is drapeability, which grew from ideas of J. R. Alexander in early 2000s defining an arc drapeable if it lies in the convex hull...

The Napoleon–Barlotti theorem belongs to the family of theorems related to the Petr–Douglas–Neumann theorem. The Napoleon–Barlotti theorem states: On the sides of an affine-regular n-gon construct regular n-gons. Then the cent...

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contai...

It was known in antiquity that the sum of the three angles of a triangle equals $\pi$. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of the four solid...

In this paper, we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin and are normalized to be one at the origin. The conjecture essentially divides th...

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic pro...

In the present paper, we give a simple proof of the Chen–Ricci inequality for submanifolds in Riemannian and Lorentzian space forms, respectively. Moreover, we extend the Chen–Ricci inequality to submanifolds in Lorentzian manifolds with...

Let $▵ABC$ be a triangle in the plane $\mathcal{E}$, K be its symmedian point, and $\mathcal{C}$ be its circumcircle. Assume that P is a point on $\mathcal{C}$ such that it is not A, B, or C, it does not lie on the medians of $▵ABC$, and the lines ${\ell }_{AP}$, ${\ell }_{BP}$, and $T_{}$...

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for $p=2$.

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in $\mathbb{R}$, the Lapidus conjecture has been confirmed. However, i...

Given an ellipse rolling along a straight line without slipping and a point P on the ellipse, we will determine the shape of the elliptic cycloid traced by P as the ellipse rolls and compute the area under one arch of the elliptic cycloid. We also in...