Pitot's theorem, dynamic geometry and conics

It is well known that a convex quadrilateral is a cyclic quadrilateral if, and only if, the sum of each pair of opposite angles is π. This result (which gives a necessary and sufficient condition for the existence of a circle which circumscribes a given quadrilateral) is beautifully complemented by Pitot’s theorem which says that a given convex quadrilateral has an inscribed circle if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair. Henri Pitot, a French engineer, noticed the easy part of this result in 1725 (see Figure 1), and the converse was first proved by J-B Durrande in 1815. Accordingly, we shall say that a convex quadrilateral is a Pitot quadrilateral if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair.

Review of Forbidden Configurations in Discrete Geometry by David Eppstein

In 1930, the mathematician Esther Klein observed that any five points in the plane in general position (i.e., no three points forming a line) contain four points forming a convex quadrilateral. This innocentsounding discovery led to major lines of research in discrete geometry. Klein's friends Paul Erdős and George Szekeres generalized this theorem, and also conjectured that 2k-2 + 1 points (again in general position) would be enough to force a convex k-gon to exist. The resolution of this conjecture became known as the "happy ending problem," because Klein and Szekeres ended up getting married. The unhappy side is that it has, to date, not been completely solved, although a recent breakthrough of Suk made significant progress. This both mathematically and personally charming little story is a great beginning for this elegant book about discrete geometry. It typifies the type of problems that are studied throughout, and also captures the spirit of curiosity that drives such studies. The book covers many problems that lie at the intersection of three fields: discrete geometry, algorithms and computational complexity.

Generalized Regularity and the Symmetry of Branches of ''Botanological'' Networks

We derive the generalized regularity of convex quadrilaterals in R^2, which gives a new evolutionary class of convex quadrilaterals that we call generalized regular quadrilaterals in R^2. The property of generalized regularity states that the Simpson line defined by the two Steiner points passes through the corresponding Fermat-Torricelli point of the same convex quadrilateral. We prove that a class of generalized regular convex quadrilaterals consists of convex quadrilaterals, such that their two opposite sides are parallel. We solve the problem of vertical evolution of a ''botanological'' thumb (a two way communication weighted network) w.r to a boundary rectangle in R^2 having two roots,two branches and without having a main branch, by applying the property of generalized regularity of weighted rectangles. We show that the two branches have equal weights and the two roots have equal weights, if the thumb inherits a symmetry w.r to the midperpendicular line of the two opposite sides of the rectangle, which is perpendicular to the ground (equal branches and equal roots). The geometric, rotational and dynamic plasticity of weighted networks for boundary generalized regular tetrahedra and weighted regular tetrahedra lead to the creation of ''botanological'' thumbs and ''botanological'' networks (with a main branch) having symmetrical branches

PREPARING FOR MATHEMATICAL COMPETITIONS: A PROBLEM SERIES ON METRIC RATIOS IN A QUADRANGLE

Solving of competitive problems by pupils and students is a good foundation and preparation for future practical and scientific activities, as mastering the methods of solving competitive problems requires them to work hard, actively and focused, as well as develops their creativity and raises level of interest in mathematics. The article reveals the mathematical aspects of preparing students to solve competitive problems on the example of one geometric problem (the ratio between the areas of triangles formed by the intersection of diagonals of a convex quadrilateral), which is the basis of many competitive problems in geometry; the problem is solved using the facts of elementary mathematics, available to students of the eighth form of secondary school; an analysis of the range of competitive problems of various mathematical competitions, for which the considered reference problem is a key subtask in the solution. An author's competitive problem for high school students has been created, which allows integrating a purely theoretical-numerical problem into the geometric shell with the study of simplicity of elements, divisibility of a product by a prime number, mutual simplicity of elements, with the need to find solutions of Diophantine equations in natural numbers. The article combines a problem series of a large number of different competitive geometric problems around one reference problem, presents the methodological aspects of preparing students to solve competitive problems on the example of this problem; attention is paid to checking the correctness of the obtained results, which avoids erroneous solutions; the tasks which urge to find and realize ways of their fulfillment are analyzed; examples of different tasks in terms of age capabilities of researchers are selected; the problems of competitions of regional levels with geometric and theoretical-numerical filling are considered; the competitive task on the given subject is created. Further research will be aimed at creating a broader series of tasks for the considered reference problem, including problems with integration into related competitive topics. The article emphasizes the problem content and structuring according to the age capabilities of students on the research topic.

New proofs of certain characterisations of cyclic circumscriptible quadrilaterals

A convex quadrilateral ABCD is called circumscriptible or tangential if it admits an incircle, i.e. a circle that touches all of its sides. A typical circumscriptible quadrilateral is depicted in Figure 1, where the incircle of ABCD touches the sides at, Aʹ, Bʹ, Cʹ and Dʹ. Notice that labellings such as AAʹ = ADʹ = a are justified by the fact that two tangents from a point to a circle have equal lengths (a, b, c and d in Figure 1 are called tangent lengths). This simple fact also implies that if x, y, and z are the angles shown in the figure, then x = y. In fact, if AD and BC are parallel, then x = y = 90°. Otherwise, the extensions of AD and BC would meet, say at Q, with QDʹ. Hence x = y. Thus x = y in all cases, and sin x = sin y = sin z. We shall use this observation freely. Also we shall denote the vertices and vertex angles of a polygon by the same letters, but after making sure that no confusion may arise.

Fooled by Rounding

For any convex quadrilateral, joining each vertex to the mid-point of the next-but-one edge in a clockwise direction produces an inner quadrilateral (as does doing so in a counter-clockwise direction). In many cases, a dynamic geometry measurement of the ratio of the area of the outer quadrilateral to the area of the inner one appears to be 5:1. It turns out, however, that this is due to rounding. We generalise the construction by replacing mid-points by more general ratios, finding the maximum and minimum values of the area ratio and determining the conditions on the original quadrilateral that achieve those two extremes.

Pascal-points quadrilaterals inscribed in a cyclic quadrilateral

This paper presents some new theorems about the Pascal points of a quadrilateral. We shall begin by explaining what these are.Let ABCD be a convex quadrilateral, with AC and BD intersecting at E and DA and CB intersecting at F. Let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. By using Pascal’s theorem for the crossed hexagons EKNFML and EKMFNL and which are circumscribed by ω, the following results can be proved [1]:– (a)NK, ML and AB are concurrent (at a point P internal to AB)(b)NL, KM and CD are concurrent (at a point Q internal to CD)