Related papers: Conics associated with triangles, or how Poncelet …
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
Three circles define each of the Brocard points of a triangle. If one adds the three circles through a pair of vertices and the orthocentre one has nine circles. It is described how each of the nine centres of these circles lies at the…
For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2\pi$. As an easy consequence of this, a proof of Euclid's…
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction…
We show how the Eulcidean algorithm for polynomials can be used to find the intersection points, with multiplicities, of two plane algebraic curves.
We consider the following configuration. Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, and for each vertex $X$, let $H_X$ be the orthocenter of the triangle formed by the other three. Then…
We establish a simple generalization for the famous theorem of Morley about trisectors in triangle with a purely synthetic proof using only angle chasing and similar triangles. Furthermore, based on the converse construction, another simple…
It is shown in this paper that non-conforming finite elements on the triangle using $P^{1}$-nonconforming polynomials and $P^{2}$ -conforming polynomials can be easily built and used.They appear as an 'enriched' version of the standard…
In a space-time, a conformal structure is defined by the distribution of light-cones. Geodesics are traced by freely falling particles, and the collection of all unparameterized geodesics determines the projective structure of the…
Trisecting an angle has been proved to be impossible by Euclidean Geometry, using only straight edge and compass. However, there is a method using Origami (paper folding) procedure to trisect an angle. The algebraic analysis of the same…
One of the most important problems in Geometric Tomography is to establish properties of a given convex body if we know some properties over its sections or its projections. There are many interesting and deep results that provide…
This paper is devoted to presenting a new approach to determine the intersection of two quadrics based on the detailed analysis of its projection in the plane (the so called cutcurve) allowing to perform the corresponding lifting correctly.…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.
It is conjectured that if a finite set of points in the plane contains many collinear triples then there is some structure in the set. We are going to show that under some combinatorial conditions such pointsets contain special…
Motivated by questions occuring in the construction of certain twistor spaces the parameter space of conics tangent to a given quartic is investigated. For a given real quartic surface in complex $\PP ^3$ that has exactly 13 ordinary nodes…
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…
In this work, we introduce a new geometry based on the difference angle, an angle defined as the difference of slopes of two lines, together with an axiomatic system for angles. This framework provides a constructive approach to the…
Morley's Theorem about angle trisectors can be viewed as the statement that a certain diagram `exists', meaning that triangles of prescribed shapes meet in a prescribed pattern. This diagram is the case n=3 of a class of diagrams we call…
We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This…