There are many problems whose solution requires proof that a quadrilateral is cyclic. The main reason for writing this paper is to offer a number of new tools for proving that a particular quadrilateral is cyclic, thus expanding the present knowledge base and ensuring that investigators in mathematics and teachers of mathematics have at their…

We use the hyperbolic cotangent function to deduce another proof of Euler's formula for ?(2n).

In this paper, we use geometric transformations to find some interesting properties related with geometric loci. In particular, given a triangle or a cyclic quadrilateral, the locus generated by the centroid or by the orthocentre (for triangles) or by the anticentre (for cyclic quadrilaterals) when one vertex moves on the circumcircle of the…

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs…

For a given positive integer k [not equal] 4, let "P[subscript k,n]" denote the "n"-th "k"-gonal number. We study "k"-gonal triples ("a", "b", "c") satisfying P[subscript k,a] + P[subscript k,b] = P[subscript k,c]. A "k"-gonal triple corresponds to a rational point on the rectangular hyperboloid x[squared] + y[squared] = z[squared] + 1. The simple…

Constructing formulas "from scratch" for calculating geometric measurements of shapes--for example, the area of a triangle--involves reasoning deductively and drawing connections between different methods (Usnick, Lamphere, and Bright 1992). Visual and manipulative models also play a role in helping students understand the underlying…

A well known result due to Laplace states the equivalence between two different ways of defining the determinant of a square matrix. We give here a short proof of this result, in a form that can be presented, in our opinion, at any level of undergraduate studies.

An outbox of a quadrilateral is a rectangle such that each vertex of the given quadrilateral lies on one side of the rectangle and different vertices lie on different sides. We first investigate those quadrilaterals whose every outbox is a square. Next, we consider the maximal outboxes of rectangles and those quadrilaterals with perpendicular…

The inter-derivability of the Pythagorean Theorem and Heron's area formula is explained by applying Al-Karkhi's factorization to Heron's formula.

Pompeiu's theorem states that if ABC is an "equilateral" triangle and M a point in its plane, then MA, MB, and MC form a new triangle. In this article, we have a new look at this theorem in the realm of arbitrary triangles. We discover what we call Pompeiu's Area Formula, a neat equality relating areas of triangles determined by the points A, B,…

We present a concise, yet self-contained module for teaching the notion of area and the Fundamental Theorem of Calculus for different groups of students. This module contains two different levels of rigour, depending on the class it used for. It also incorporates a technological component. (Contains 6 figures.)

In this article, we investigate the construction of spirals on an equilateral triangle and prove that these spirals are geometric. In further analysing these spirals we show that both the male (straight line segments) and female (curves) forms of the spiral exhibit exactly the same growth ratios and that these growth ratios are constant…

The author presents a refinement of the Cauchy-Schwarz inequality. He shows his computations in which refinements of the triangle inequality and its reverse inequality are obtained for nonzero x and y in a normed linear space.

Learning about the area formulas provides many opportunities for students even at the beginning of junior secondary school to experience mathematical deduction. For example, in easy cases, students can put two triangles together to make a rectangle, and so deduce that the area of a triangle is half the area of a corresponding rectangle. They can…

The basic Pythagorean theorem for right-angled triangles is well-known in mathematical terms as a[squared]+b[squared]+c[squared] were "a," "b," and "c" are the lengths of the sides of the triangle with "c" as the hypotenuse. When "a," "b," and "c" are all integers and obey this…