Related papers: Parabolic polygons
We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by M\"obius, using hyperbolic geometry.
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits…
Given a plane oval, is it possible to go around it so that, at all times, the two tangent segments to the oval from the moving point have unequal lengths? In this note we construct an example of such an oval.
The Miquel-Steiner theorem for a quadrilateral in the Euclidean plane states that the circumcircles of the four component triangles intersect at a single point, which now is called the Miquel-Steiner point of the quadrilateral. In elliptic…
We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erd\H{o}s.
In this pedagogical note we present a short proof of the following main result of arxiv.org/abs/0911.5319, and clarify its relation to the isoperimetric problem. On the hyperbolic plane consider triangles ABC with fixed lengths of AB and…
In this note, we will explain the connection between the Seven Circles Theorem and hyperbolic geometry, then prove a stronger result about hyperbolic geometry hexagons which implies the Seven Circles Theorem as a special case.
We consider the family of cubic polynomials with a simple parabolic fixed point. We prove that the boundary of the immediate basin of attraction of the parabolic point is a Jordan curve and give a description of the dynamics.
In this work the confined domains for a point-like particle propagating within the boundary of an ideally reflecting paraboloid mirror are derived. Thereby it is proven that all consecutive flight parabola foci points lie on the surface of…
We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled…
Let M be a smooth strictly convex closed surface in space and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface…
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…
The celebrated theorem of Feuerbach states that the nine-point circle of a nonequilateral triangle is tangent to both its incircle and its three excircles. In this note, we give a simple proof of Feuerbach's Theorem using straightforward…
Let $\mathcal{P}$ be a set of $n$ points in the Euclidean plane. We prove that, for any $\epsilon > 0$, either a single line or circle contains $n/2$ points of $\mathcal{P}$, or the number of distinct perpendicular bisectors determined by…
A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viwed as the converse of Tarski's plank problem (Bang's theorem): Is it true that if the total width of a collection of planks…
Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord $AB$ on the parabola, let us denote by $P$ the point on the parabola where the tangent is parallel to…
In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points…
Part I: The two-dimensional Pascal Triangle will be generalized into a three-dimensional Pascal Pyramid and four-, five- or whatsoever-dimensional hyper-pyramids. Part II: The Bilateral Binomial Theorem will be generalised into a Bilateral…
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational…