Related papers: Bisecting a triangle in a given direction
A long-standing, unanswered question regarding Euclid's Elements concerns the absence of a theorem for the concurrence of the altitudes of a triangle, and the possible reasons for this omission. In the centuries following Euclid, a…
We show that for certain triangulations of surfaces, circle packings realising the triangulation can be found by solving a system of polynomial equations. We also present a similar system of equations for unbranched circle packings. The…
If a line cuts randomly two sides of a triangle, the length of the segment determined by the points of intersection is also random. The object of this study, applied to a particular case, is to calculate the probability that the length of…
Heron's formula states that the area $K$ of a triangle with sides $a$, $b$, and $c$ is given by $$ K=\sqrt {s(s-a) (s-b) (s-c)} $$ where $s$ is the semiperimeter $(a+b+c)/2$. Brahmagupta, Robbins, Roskies, and Maley generalized this formula…
For a given triangle $\triangle ABC$, we define two sequences of points on line $BC$ and provide their generalizations to real functions such that centers of circumscribed circles around $A$ and adjacent points in subsequences generate a…
This paper explores and proves the one-seventh area triangle using a purely algebraic approach as opposed to a geometric one. A triangle set purely in the complex plane is used so that we can utilise features of the complex number system to…
We consider the following problem: Let $\mathcal{L}$ be an arrangement of $n$ lines in $\mathbb{R}^3$ colored red, green, and blue. Does there exist a vertical plane $P$ such that a line on $P$ simultaneously bisects all three classes of…
Visibility graph of a simple polygon is a graph with the same vertex set in which there is an edge between a pair of vertices if and only if the segment through them lies completely inside the polygon. Each pair of adjacent vertices on the…
A complete treatment of the intersections of two geodesics on the surface of an ellipsoid of revolution is given. With a suitable metric for the distances between intersections, bounds are placed on their spacing. This leads to fast and…
We study the relationship between the areas of the consecutive quadrilaterals cut from a convex quadrilateral in the plane by means of a finite or infinite number of straight lines intersecting two of its opposite sides. Moreover, we obtain…
The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank…
We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…
We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane.
The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…
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.
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…
Sweeping is a commonly used procedure to explicitly solve the discrete ordinates equation, which itself is a common approximation of the neutron transport equation. To sweep through the computational domain, an ordering of the spatial cells…
A closed-form solution for the boundary of the flat state of an orthogonal cross section of contiguous surface geometry formed by the intersection of two cylinders of equal radii oriented in dual directions of rotation about their…
Three linearly dependent and pairwise linearly independent vectors of an euclidian space uniquely determine a planar quadric with symmetry centre in the origin. A rather simple formula for the area of an arbitrary sector at centre of such a…
Se enuncia los principales teoremas empleados en la resoluci'on de tri'angulos oblicu'angulos. Con ellos, se ilustra c'omo resolver los cinco casos de resoluci'on que se presentan, incluyendo algunos caso at'ipicos (cuando se conoce el…