The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal, then the triangle is isosceles. In other words, if P is the incentre of triangle ABC, and if BP and CP meet the sides AC and AB at B′and C′, respectively, then

The human mind, by nature, is curious and enjoys dealing, both independently and competitively, with intellectual challenges. Throughout time, mathematical tasks, riddles, and puzzles have offered such challenges. In today's modern era, marketing companies and the media even offer prizes and rewards for the successful solution of puzzles, expanding the audience exposed to the various challenges. In addition, the styles, subjects and depths of the puzzles have become diversified. Experience gained in solving puzzles aids in developing reasoning abilities and deepening thought.

The purpose of this paper is to collect for the first time area formulas for all convex quadrilaterals that can be expressed in terms of their four sides alone. There are presently 23 such classes that we know of. Many mathematics teachers would probably find it challenging just to name that many different classes of quadrilaterals, and even more difficult to tell how to calculate their different areas. Several of these classes of quadrilaterals are included neither in MathWorld, Wikipedia nor in any other encyclopedia that we know of. The formulas can be sorted into four categories as either well known, a special case of another formula, less known, or previously unknown (at least to us). The last category only includes two formulas and we will prove these.

There is a long tradition of using geometry to solve problems from mechanics. Unfortunately this tradition is not practised much in schools and university any more.

With this exposition we would like to demonstrate how elementary properties of ellipses can be used to solve a problem related to the conical pendulum. The problem of the conical pendulum is to consider a mass attached to one end of a light inextensible string of length with the other end attached at the top of a vertical rod. The mass is moving about the rod in uniform circular motion in a horizontal plane. Given the angular velocity of the mass, the question is to determine the angle the string makes with the rod.

Lack of information is generally considered a hindrance to inquiry. Surprisingly, a simple mathematical argument, relying on the Principle of Indifference, shows there are situations where the opposite holds. Even more surprisingly, this indifference allows one to guess, with a success rate greater than 50%, the outcome of a coin toss or any other experiment having two equiprobable outcomes. The scheme is based on work by American statistician David Blackwell (1919–2010) and a principle of mathematical probability attributed to Swiss mathematician Jakob Bernoulli (1655–1705).

A number of methods are available for the solution of cubic equations. Amongst these, and possibly less well known than those based on algebraic or trigonometrical procedures, is an elegant graphical solution devised by Lalanne [1]. His approach and the resulting nomogram are appraised in this paper.

The general cubic equation in the variable X

Spirals are among the most intriguing of geometrical patterns, frequently encountered in the world around us at all scales, from the cosmic spiral arms of galaxies to the microscopic structure of the DNA molecule. For centuries, humans have studied these patterns, classifying them, giving them different names, and describing them mathematically. The most common definition of a two-dimensional spiral is a curve on the plane traced by a point that winds around a certain fixed point (called the spiral's pole), while monotonically approaching or receding from it, depending on the direction of motion.

While the notion of roots of a quadratic polynomial is rudimentary in high school mathematics, that of its fixed points is uncommon. A real or complex number is a fixed point of a polynomial p (x) p (θ) = θ. The fact that the notion of fixed point of polynomials is not commonly covered in high school or undergraduate mathematics is surprising because the relevance of the fixed points of a quadratic can be demonstrated easily via iterative methods for the approximation of such numbers as , when the quadratic formula offers no remedy.

We shall present some new proofs concerning the distribution of the digits in decimals. We finish with a proof of Borel's normal number theorem. The proof given here is more direct and straightforward than the one given in [1]. This theorem is concerned with the distribution of the digits within infinite decimals. The actual statement of the theorem will be given later after some intermediate results have been established. The theorem is not only unexpected but seemingly counter-intuitive. In order to follow the arguments given here it is useful to have a numerical example as we will count the decimals and then their digits. We shall deduce the results for a general base b.

In this Article, we derive a method for classifying all positive integer solutions of the equation x2 + ry2 = z2, where r is a given positive rational. In order to simplify notation, such a solution with x = a, y = b and z = c will be denoted by the ordered triple (a, b, c) and, in all that follows, the term solution will be taken to mean a positive integer solution of the above equation. The method employed will be similar to that used in [1] to find all integer triangles containing an angle whose cosine is known.

In a previous paper [1], I described a method for computing sums of powers via a matrix composed from Pascal's triangle. (The motivation for this came from a mathematical puzzle posed by Paul Maslanka in the Guardian.) Since then I have read or viewed a number of papers on sums of powers. Some were nearly four hundred years old and the most recent one was published this year. One rather obscure work I stumbled across was ‘On the sums of powers of natural numbers’ by the Japanese mathematician Mr T. Kariya, published in Tokyo in 1907. If any readers know more of Mr Kariya I would be glad to hear about him.

There are recursive expressions (see [1]) for sequentially generating the integer solutions to Pell's equation: p2 − Dq2 = 1, where D is any positive non-square integer. With known positive integer solution p1 and q1 we can compute, using these recursive expressions, pn and qn for all n > 1. See Table in [2] for a list of smallest integer, or fundamental, solutions p1 and q1 for D≤ 128. These (pn, qn) pairs also form rational approximations to that, as noted in [3, Chapter 3], match with convergents (Cn = pn / qn) of the Regular Continued Fractions (RCF, continued fractions with the numerator of all fractions equal to 1) for .

The number πis perhaps the most famous irrational number. This constant is equal to the ratio of the circumference of a circle to its diameter. One of the most well-known mathematical problems of antiquity, which is related to π, is how to construct by using a ruler and compasses a square which has the same area as a circle. This particular problem cannot be solved, due to the fact that π is a transcendental number, which means that it cannot be obtained as the root of a polynomial equation with rational coefficients. It was Euler in the 18th century who established the notation π.

This Article is on the discrete Fourier transform (DFT) and the fast Fourier transform (FFT). As we shall see, FFT is a slight misnomer, causing confusion to beginners. The idiosyncratic title will be clarified in §4.

Computing machines are highly efficient nowadays, and much of the efficiency is based on the use of the FFT to speed up calculations in ultrahigh precision arithmetic. The algorithm is now an indispensable tool for solving problems that involve a large amount of computation, resulting in many useful and important applications: for example, in signal processing, data compression and photo-images in general, and WiFi, mobile phones, CT scanners and MR imaging in particular.