Related papers: A New Equilateral Triangle?
We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite…
In this paper, we present a synthetic solution to a geometric open problem involving the radical axis of two strangely defined circumcircles. The solution encapsulates two generalizations, one of which uses a powerful projective result…
In 1840 Jacob Steiner on Christian Rudolf's request proved that a triangle with two equal bisectors is isosceles. But what about changing the bisectors to cevians? Cevian is any line segment in a triangle with one endpoint on a vertex of…
We define spherical Heron triangles (spherical triangles with "rational" side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many…
The incircle of a triangle touches the sides of the triangle in three points. It is well known that the lines from these points to the opposite vertices meet at a point known as the Gergonne point of the triangle. We use a computer to…
In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study.…
An upward equilateral triangle of side $n$ can be partitioned into $n$ unit upward equilateral triangles and $\frac{n(n-1)}{2}$ unit rhombi with $60^{\circ}$ and $120^{\circ}$ angles. In this paper, we focus on understanding such partitions…
We classify perimeter dominant triangles whose side lengths are in $\sqrt3\mathbb N$ and whose area is in $\frac{\sqrt3}4\mathbb N$. There is one exceptional example, which is equilateral, and three infinite families determined by certain…
Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into…
We suggest a geometric visualization of the process of constructing a triangle with prescribed bisectors that makes the existence of such a triangle geometrically evident.
A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the…
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of…
We analyze loci of triangle centers over variants of two-well known triangle porisms: the bicentric and confocal families. Specifically, we evoke the general version of Poncelet's closure theorem whereby individual sides can be made tangent…
We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.
A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if $\Delta$ is a perfect triangle, then it cannot be an isosceles triangle. It gives a…
We introduce a linear algebraic object called a bidiagonal triple. A bidiagonal triple consists of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the…
Percolation on a plane is usually associated with clusters spanning two opposite sides of a rectangular system. Here we investigate three-leg clusters generated on a square lattice and spanning the three sides of equilateral triangles. If…
We determine the conditions resulting from equating the area sums of alternative sectors in a circle generated by four, two, and three straight lines, respectively, that connect opposite points on its circumference while passing through a…
We prove that every solution of the Helmholtz equation within an equilateral triangle, which obeys the Dirichlet conditions on the boundary, is a member of one of four symmetry classes. We then show how solutions with different symmetries,…
We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants…