Related papers: More Shapes of Central Quadrilaterals
The diagonals of a quadrilateral form four component triangles (in two ways). For each of various shaped quadrilaterals, we examine 1000 triangle centers located in these four component triangles. Using a computer, we determine when the…
The diagonals of a quadrilateral form four associated triangles, called half triangles. Each half triangle is bounded by two sides of the quadrilateral and one diagonal. If we locate a triangle center (such as the incenter, centroid,…
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…
We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of…
The circumcircle of a planar convex polygon P is a circle C that passes through all vertices of P. If such a C exists, then P is said to be cyclic. Fix C to have unit radius. While any two angles of a uniform cyclic triangle are negatively…
The four Hagge circles of the triangles BCD, ACD, ABD, ABC of a cyclic quadrilateral ABCD with respect to an appropriate common axis of inverse similarity share many features which are analysed.
If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of…
We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from…
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…
This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also…
We revisit constructions based on triads of conics with foci at pairs of vertices of a reference triangle. We find that their 6 vertices lie on well-known conics, whose type we analyze. We give conditions for these to be circles and/or…
An equilateral pentagon is a polygon in the plane with five sides of equal length. In this paper we classify the central configurations of the $5$-body problem having the five bodies at the vertices of an equilateral pentagon with an axis…
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.…
We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans…
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…
Among a triangle's exparabolas (parabolas escribed to the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's…
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…
A rhombus tiling of a hexagon is said to be centered if it contains the central lozenge. We compute the number of vertically symmetric rhombus tilings of a hexagon with side lengths $a, b, a, a, b, a$ which are centered. When $a$ is odd and…
Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections.…
The central component of a polygon triangulation is defined as the triangle or diameter that contain its geometric center. More generally, every polygon dissection contains a central component. Using this notion, we derive new recurrences…