Related papers: A little noticed right triangle
Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using…
Octonion algebras over rings are, in contrast to those over fields, not determined by their norm forms. Octonion algebras whose norm is isometric to the norm q of a given algebra C are twisted forms of C by means of the Aut(C)-torsor O(q)…
Non-Euclidean triangle centers can be described using homogeneous coordinates that are proportional to the generalized sines of the directed distances of a given center from the edges of the reference triangle. Identical homogeneous…
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…
In this paper, we derive the exact formula for the probability that three randomly and uniformly selected points from the interior of the unit cube form vertices of an obtuse triangle.
In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given…
The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are…
For an arbitrary convex quadrilateral $ABCD$ with area ${\cal A}$ and perimeter $p$, we define two points $I_1, I_2$ on its Newton line that serve as incenters. These points are the centers of two circles with radii $r_1, r_2$ that are…
A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…
We suggest a method of solving the problem of existence of a triangle with prescribed two bisectors and one third element which can be taken as one of the angles, the sides, the heights or the medians, or the third bisector.
We present an intriguing question about lattice points in triangles where Pick's formula is "almost correct". The question has its origin in knot theory, but its statement is purely combinatorial. After more than 30 years the topological…
We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude,…
Let ABC be a triangle with a,b,and c being its three sidelengths. In a 1976 article by Wynne William Wilson in the Mathematical Gazette(see reference[2]), the author showed that angleB is twice angleA, if and only if b^2=a(a+c). We offer…
Let P be a point inside a convex quadrilateral ABCD. The lines from P to the vertices of the quadrilateral divide the quadrilateral into four triangles. If we locate a triangle center in each of these triangles, the four triangle centers…
An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This…
We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning…
In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers…
Given a triangle, what is the equation of the line which bisects its area and has a given slope? The set of all lines bisecting the area of a triangle has been elegantly determined as a certain 'deltoid' envelope and this gives an indirect…
A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then…
A well-known object in classical Euclidean geometry is the circumcenter of a triangle, i.e., the point that is equidistant from all vertices. The purpose of this paper is to provide a systematic study of the circumcenter of sets containing…