Related papers: Relationships Between Six Circumcircles

This paper concerns the number of lattice points in a circle.

Two polygons, $(P_1,\ldots,P_n)$ and $(Q_1,\ldots,Q_n)$ in ${\mathbb R}^2$ are $c$-related if $\det(P_i, P_{i+1})=\det(Q_i, Q_{i+1})$ and $\det(P_i, Q_i)=c$ for all $i$. This relation extends to twisted polygons (polygons with monodromy),…

We address the claim of Weiss et al. (2018) that the radii of adjacent planets in Kepler multi-planet systems are correlated. We explore two simple toy models---in the first the radii of the planets are chosen at random from a single…

An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. We wish to understand…

We show that if a disc triangulation has all internal vertex degrees at least 6, then the full triangulation may be determined from the pairwise graph distance between boundary vertices. A similar result holds for quadrangulations with all…

Let $\mathcal{K}$ denote a nonsingular conic in the complex projective plane. Pascal's theorem says that, given six distinct points $A,B,C,D,E,F$ on $\mathcal{K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are…

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As…

Given a triangle ABC, we derive the probability distribution function and the moments of the area of an inscribed triangle RST whose vertices are uniformly distributed on AB, BC, and CA. The theoretical results are confirmed by a Monte…

In this work I look at the distribution of primes by calculation of an infinite number of intersections. For this I use the set of all numbers which are not elements of a certain times table in each case. I am able to show that it exists a…

In any triangle, the perpendicular side bisectors meet the corresponding internal angle bisectors on the circumcircle. If we take those three points as the vertices of a new triangle and repeat the operation indefinitly, we end up in the…

We determine the distribution of nearest neighbour spacings between the tangencies to a fixed circle in a class of circle packings generated by reflections. We use a combination of geometric tools and the theory of automorphic forms.

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

Main Theorem. Two parabols have four common points. There exists a circle tangent to the sides of the obtained parabolic quadrilateral if and only if the diagonals of this quadrilateral are orthogonal. The proof of the Main Theorem is…

We consider the Poisson cylinder model in ${\mathbb R}^d$, $d\ge 3$. We show that given any two cylinders ${\mathfrak c}_1$ and ${\mathfrak c}_2$ in the process, there is a sequence of at most $d-2$ other cylinders creating a connection…

In this paper we investigate the relationships between envelopes of circle families and some special curves in the plane, such as evolutes, pedals, evolutoids and pedaloids.

In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle $ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the lines $AA_1,…

The topology of the intersection of three quadrics in Euclidean 6-space is studied using Kollar results. This needs an existence of a line without real points in the complex projectivisation of quadrics. We establish the existence of such a…

Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R.…

We start with certain joint densities (for sides and for angles) corresponding to pinned Poissonian triangles in the plane, then discuss analogous results for staked and anchored triangles.

We derive some, seemingly new, curious additive relations in the Pascal triangle. They arise in summing up the numbers in the triangle along some vertical line up to some place.