USING THE DUALITY PRINCIPLE WHEN CONSTRUCTING EXERCISES AT THE GEOMETRY LESSONS

Geometry is one of the complex disciplines where many facts are interconnected. It is possible to develop the idea of facts interrelations through correlation using the duality principle. The duality principle is known in projective geometry, mathematical logic. This principle is clearly pronounced in one of the theorems of new triangle geometry. The traditional analytical geometry course does not study the facts of new triangle geometry. To reinforce many topics of the analytical geometry course, for example, “The distance between two points”, “The symmetrical form of the equation of a line”, “The angle between two lines”, it is reasonable to consider some facts from the new triangle geometry in the Cartesian coordinate system. Thus, an element of novelty is introduced to the reviewed material. The guidebooks on triangle geometry solve tasks through classical approaches or applying barycentric coordinates not using analytical geometry formulas. The paper proposes the constructing technique for the couples of exercises using the duality principle in the plane geometry teaching methods. Tasks are constructed for the Cartesian coordinate system as this allows demonstrating the duality of points in the drawings. In the composed exercises, two drawings are constructed in parallel columns. In different cases, the points can be the triangle-apexes, an orthocenter, or a height base. The initial triangle sides are located on the axes of coordinates, and their side lengths set up Pythagorean triple for better understanding the task-solving algorithm by the students. The symmetrical form of the equation of a line shows the necessity of analytical study since it is difficult to check the distance from the orthocenter to the orthotriangle sides in the drawings due to the small value. For many such information units, the aggregation relationships (whole-part) are set up, reflecting the geometric embedding of components.

Eight formulae for the area of triangle OIH

Which is your favourite formula in triangle geometry? Mine is definitely the formula for the area of triangle OIH . It is well known that the perpendicular bisectors to the sides of any triangle are concurrent at a point O (centre of the circumcircle), that the angle bisectors to the vertex angles are concurrent at a point I (centre of the incircle), and that the altitudes are concurrent at a point H. If the triangle is not isosceles, then these three points are all different and uniquely determine a new triangle OIH (see Figure 1), whose area can be expressed in terms of the sides a, b, c of the original triangle. I derived such a formula 20 years ago, and later found out that it had been studied a century earlier.

The mechanism of hairiness reduction in offset ring spinning with a diagonal yarn path

Reducing yarn hairiness on a ring spinning machine generally involves an investment and ongoing process cost. Although conceptually successful, the cost factor associated with existing technologies has led to exploration for new ways of reducing hairiness. Offset spinning is an outcome of one such novel initiative in which the spinning triangle geometry is altered by diagonally offsetting the yarn path during spinning. This technique has been reported to reduce hairiness but a consensus on a specific direction of offset (left or right) remains pending as its underlying mechanism is still not completely understood. In this study, we developed an experimental setup to image a spinning triangle geometry in detail and observed the effect of offsetting its shape on fiber path. It was found that the direction of offset (left or right) in combination with a specific twist direction (S or Z) can result in a complete change of yarn twist configuration (Archimedean or Fermat's spiral), which ultimately controls hair generation tendency during spinning. On the basis of agreement between imaging observations and hairiness parameter results obtained from yarns spun with different offset conditions, a mechanism of hairiness reduction and preferred direction of offset is proposed.