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In this note, primarily intended for high school students and high school teachers, characterizations of a right triangle and an equilateral triangle in the Euclidean plane are presented using the nine-point circle of a given triangle. Geometrical applications are explored along with their possible uses in the teaching environment. (Contains 4 figures.)

Three unheralded results, two in Coordinate Geometry and the other in Plane Geometry, provide three proofs of a theorem dubbed by this author as "The Fundamental Theorem". The above theorem offers a uniform proof establishing the positive integral nature of the radii of the incircle and the three escribed circles associated with the right triangle corresponding to any Pythagorean triplet of natural numbers. Another special selection of Pythagorean triplet ensures the positive integral nature of the radii of six circles namely; the incircle, circumscribing circle, the three escribed circles and the nine point circle of the right triangle that comes from the specialized Pythagorean triplet. Through these connections some other new results that can be deduced from the Pythagorean triplet are presented in this article. (Contains 2 figures.)

This article presents different approaches to a problem, dubbed by the author as "the consecutive pages problem". The aim of this teaching-oriented article is to promote the teaching of abstract concepts in mathematics, by selecting a challenging amusement problem and then presenting various solutions in such a way that it can engage the attention of a fourth-grade student, a high school senior student, an average college student and scholars.

The purpose of this article is to discuss specific techniques for the computation of the volume of a tetrahedron. A few of them are taught in the undergraduate multivariable calculus courses. Few of them are found in text books on coordinate geometry and synthetic solid geometry. This article gathers many of these techniques so as to constitute a minor survey of a teaching-oriented article, useful to both students and teachers according to their needs in the classrooms.

The author introduces the birthday problem in this article. This can amuse willing members of any birthday party. This problem can also be used as the motivational first day lecture in number theory for the gifted students in high schools or in community colleges or in undergraduate classes in colleges.

Three distinct points A = (a, 0, 0) B = (0, b, 0) and (c, 0, 0) with abc not equal to 0 are taken, respectively on the "x", "y" and the "z"-axes of a rectangular coordinate system in R[superscript 3]. Using the converse of the theorem of Pythagoras, it is shown that the triangle [delta]ABC can never be a right-angled triangle. The result seems to be intuitive, but nevertheless requires a proof. As an application, some intuitive results about a tetrahedron are confirmed.

Procedural evaluations of limits of functions provide invariably better understanding of the limits than the approximations using a calculator. The purpose of this article is to demonstrate that better understanding can be promoted if mathematical understanding precedes the impulse to use calculators. The note clarifies the stages when the calculator can be used to enhance the understanding. (Contains 1 figure.)

Given a conic section, the locus of a moving point in the plane of the conic section such that the two tangent lines drawn to the conic section from the moving point are all mutually perpendicular is a curve. In the case of an ellipse and hyperbola this curve is a circle referred to as the director circle. In the case of the parabola this curve coincides with the directrix of the parabola. The last section is devoted to the graphical illustrations of director circles for circles, parabolas, ellipses and hyperbolas using the built-in Maple V software of Scientific Work Place 3.0. (Contains 4 figures.)

Given two circles C 1 and C 2 in a plane such that neither one of the two circles is contained in the other, there are either four common tangents when the circles do not intersect at all or the circles have three common tangents when they touch each other externally or only two common tangents when the circles intersect exactly at two points. The article oulines analytical procedures for computing the equations of these common tangents. Using the built-in Maple of the software Scientific Work Place 3.0, the diagrams of the circles with their common tangents are incorporated in this article. (Contains 3 figures.)

A deliberate attempt is made in Business Mathematics oriented text books as well as in some reform calculus oriented text books to interpret the derivative f[prime](a) of a function y = f(x) at the value x = a as the change in the y-value of the function per "unit" of change in the x-value. This note questions the above interpretation and suggests the necessary modification for the correct interpretation. (Contains 1 figure.)

This study aimed to explore the effect of mandibular advancement splints (MAS) on upper airway anatomy during wakefulness in obstructive sleep apnoea (OSA). Patients commencing treatment for OSA with MAS were recruited. Response to treatment was defined by a >or=50% reduction in the apnoea/hypopnoea index. Nasopharyngoscopy was performed in the supine position. Nasopharyngoscopy was performed in 18 responders and 17 nonresponders. Mandibular advancement caused an increase in the calibre of the velopharynx (mean+/- sem +40+/-10%), with relatively minor changes occurring in the oropharynx and hypopharynx. An increase in cross-sectional area of the velopharynx with mandibular advancement occurred to a greater extent in responders than nonresponders (+56+/-16% versus +22+/-13%; p<0.05). Upper airway collapse during the Müller manoeuvre, relative to the baseline cross-sectional area, was greater in nonresponders than responders in the velopharynx (-94+/-4% versus -69+/-9%; p<0.01) and oropharynx (-37+/-6% versus -16+/-3%; p<0.01). When the Müller manoeuvre was performed with mandibular advancement, airway collapse was greater in nonresponders than responders in the velopharynx (-80+/-11% versus +9+/-37%; p<0.001), oropharynx (-36+/-6% versus -20+/-5%; p<0.05) and hypopharynx (-64+/-6% versus -42+/-6%; p<0.05). These results indicate that velopharyngeal calibre is modified by MAS treatment and this may be useful for predicting treatment response.