Mediterranean Journal of Social Sciences MCSER Publishing, Rome-Italy

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1 IN 39-7 (online) IN (print) Mediterranean Journal of ocial ciences MER Publishing, Rome-Italy ol 5 No 9 ugust 4 INNTIN IN THE MTHEMTI TEHING IN ENDRY HL Djoo G Marović University of Montenegro, Faculty of Philosophy, 846 Nišić, Danila ojovic bb, Montenegro, djoogm@uninpedurs DI:59/mjss4v5n9p69 bstract The subjects of research are concrete possibilities of use philosophical-historical-developed and methodological-innovative elements in the mathematics teaching in elementary and secondary school in breaing of formalism and activating eg dynamism of teaching The aim is that the pupils thin over with originality and creation thining It develops usage of indepedent thining, critical estimation and reasonable generalization Keywords: Innovations, philosophical-historical-developed elements in the mathematics teaching, methodologicalinnovations, breaing of formalism, dynamism of teaching mathematics Introduction Properties of important points of a triangle (orthocenter, center of gravity, centers of a circumscribed circle and centers of an inscribed circle) as well as methods of their construction have been nown since ancient times However, it is less nown about the eistence, properties and structures of some unusual points of a triangle, such as J teiner s ( ) point (which has the property that the sum of the distance from it to the ape of a triangle is minimal), or H rocard s (845-9) point (with the property = = where is the inner point of triangle ), or of some other less important points n this occasion we will not deal with the well-nown properties and structures of J teiner s point, nor with the properties and structures of rocard s point of a triangle, but with properties and constructions of a less distinctive and less familiar point of a triangle for which the following theorem is true Properties of an Unusual Point of a Triangle Theorem If three congruent circumferences intersecting in point and each touching two sides of are inscribed in the triangle and if points and are the centers of the circumscribed and inscribed circles of, then points, and are collinear 69

2 IN 39-7 (online) IN (print) Proof: Mediterranean Journal of ocial ciences MER Publishing, Rome-Italy ol 5 No 9 ugust 4 Let us designate with points, and centers of the given congruent circumferences It is easy to notice that they belong to the bisectors of the interior angles in similar way, we conclude that (see figure ) L G and, from which follows that In the and have common bisectors of the interior angles, that is they have one common center of the inscribed circles It is easy to note that a homothety H with center and coefficient maps into, ie H ( ), where However, since we have that, according to the given problem, is the center of the circumscribed circle around ince and, point are homothetic, the circles circumscribed around them are also homothetic, therefore we have that H ( ), ie points, and are collinear, which was to be proven L G K (, r) Figure Property of colinearity of points, and enables us to construct point, that is to solve the following problem 3 onstructions of an unusual point of a triangle Inscribe (construct) in the given triangle three congruent circumferences intersecting in point, where each circumference touches two sides of the triangle 6

3 IN 39-7 (online) IN (print) Mediterranean Journal of ocial ciences MER Publishing, Rome-Italy ol 5 No 9 ugust 4 nalysis: ccording to the previous theorem, we have that a homothety H with center and coefficient maps into, ie H ( ), where (see figure ) In order to construct the requested point of the given triangle it is necessary and sufficient to find the homothety (similarity) coeficient For that purpose, we construct a figure similar (homothetic) to the one in picture That figure given in picture consists of a pair of homothetic triangles and which due to similarity to the figure in picture, ie similarity to a corresponding pair of homothetic triangles and have the same homothety coefficient We obtain this figure by constructing in the following way: First let us construct any which is similar to a pair of homothetic triangles and, and which will concurrently be homothetic with triangle, where the center of that homothety is point, and coefficient, ie H ( ) Let us construct centers of inscribed and circumscribed circles in and around, that is let us construct bisectors of its interior angles in whose intersection is found point - the center of the circle inscribed in, and let us construct bisectors of its sides in whose intersection is found point - the center of its circumscribed circle (see figure ) Let us construct three congruent circumferences which have one common point (the center of the circumscribed circle around ), with centers, and and with equal radiuses y It is easy to construct common tangents ), ) and ) on pairs of those circumferences that generate ( ( triangle homothetic with ( ince the homothetic figure in picture is constructed in the way that it is similar to the homothetic figure in picture, it means that the homothety coefficent is 6

4 IN 39-7 (online) IN (print) onstruction (first way): Mediterranean Journal of ocial ciences MER Publishing, Rome-Italy ol 5 No 9 ugust 4 Through analysis we found that a pair of homothetic triangles ( and ) of the figure in picture has the homothety coefficient that equals the one of a pair of corresponding homothetic triangles ( and ) of the figure in picture, ie Figure 3, therefore we can easily construct the requested segment Let us observe a part of the construction in figure, precisely let us construct a pair of corresponding homothetic triangles separately ( and ) (see figure 3) Triangle is given and we can easily construct points (the intersection of bisectors of its interior angles) and (the intersection of bisectors of its sides) (see figure 4) Homothety H with center and coefficient maps into, ie H ( ), where Therefore, in figure 4, point is to be constructed, that is the segment s we constructed a segment, we can circumscribe in figure 3 a circle K (, ) and a radius, and since center With this homothety, points, with a center in the homothety match points,,, ie (, ie is obtained as intersection of straight lines ( ) and ) H ( ) Point ( ) ( ) Thus we constructed segment requested point congruent circumferences in and on segment we easily find the of triangle In the similar way we can construct vertees of that have the common intersection in point (figure 4), and thus the three 6

5 IN 39-7 (online) IN (print) Mediterranean Journal of ocial ciences MER Publishing, Rome-Italy ol 5 No 9 ugust 4 Figure 4 onstruction of the requested point could have been done in the following way as well: Let us construct again a pair of homothetic triangles ( and ) of the figure as in picture, and then let us construct in homothety with It is clear that in this case their homologous (corresponding) sides are parallel enter of a new homothety is found in intersection of pencils of straight lines ( ) ( ) ( ) ( ), where the coefficient of this homothety (similarity) is ( ) ( ) is found in intersection of straight lines ) and ) ( ) (, then we have H and H Therefore, point the construction of point is thus completed (see figure 5) ( ( However, we have that H, and: ) (, ie ), and ( ) ( ), ( ) ( ), ( ) ( ), therefore we can construct point in a third way as a center of a circle circumscribed around 63

6 IN 39-7 (online) IN (print) Mediterranean Journal of ocial ciences MER Publishing, Rome-Italy ol 5 No 9 ugust 4 y y y y y lia 5 Proof: Proof ensues from analysis and construction Discussion: ince point as the center of the circumscribed circle around is single (only one), the problem has only one solution uch innovations presented in classes of teaching in high school result in students increased interest in independent research or creative wor and serve for eradicating formalism and activating or dynamazing the teaching process with talented students 64

7 IN 39-7 (online) IN (print) REFERENE Mediterranean Journal of ocial ciences MER Publishing, Rome-Italy ol 5 No 9 ugust 4 M ozic, (), urvey of the history and philosophy of mathematics, Institute for Tetboos and Teaching ids, elgrade G Polya, (966), How do I solve mathematical problems, school boos, Zagreb, roatia Marović, Đoo G (6) Geometrical polymorphism (Geometrijsi polimorfiza) (erbian), Nast Mat 5, No 3-4, -4 Đ G Marovic, (8), New views on methods of teaching mathematics, M3 Macarius, Podgorica 65