INNOVATIONS IN THE MATHEMATICS TEACHING IN SECONDARY SCHOOL

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 breaking of formalism and activating e.g. dynamism of teaching. The aim is that the pupils think over with originality and creation thinking. It develops usage of indepedent thinking, critical estimation and reasonable generalization.


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 known since ancient times.However, it is less known about the existence, properties and structures of some unusual points of a triangle, such as J. Steiner's (1796Steiner's ( -1863) ) point (which has the property that the sum of the distance from it to the apex of a triangle is minimal), or H. Brocard's (1845Brocard's ( -1922) ) point (with the property AB= BC= CA where is the inner point of triangle ABC), or of some other less important points.1 On this occasion we will not deal with the well-known properties and structures of J. Steiner's point, nor with the properties and structures of Brocard'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 1.If three congruent circumferences intersecting in point 1 S and each touching two sides of ABC are inscribed in the triangle and if points O and S are the centers of the circumscribed and inscribed circles of ABC , then points O , S and 1 S are collinear.

Proof:
Let us designate with points 1 1 , B A and 1 C centers of the given congruent circumferences.It is easy to notice that they belong to the bisectors of the interior angles in ABC (see figure 1).
AB In the similar way, we conclude that BC from which follows that ABC and have common bisectors of the interior angles, that is they have one common center S of the inscribed circles.It is easy to note that a homothety H with center S and coefficient k maps However, since we have that, according to the given problem, are homothetic, the circles circumscribed around them are also homothetic, therefore we have that H and S are collinear, which was to be proven.

, S O
and S enables us to construct point 1 S , that is to solve the following problem.

Constructions of an unusual point of a triangle
2. Inscribe (construct) in the given triangle ABC three congruent circumferences intersecting in point 1 S , where each circumference touches two sides of the triangle.
Analysis: According to the previous theorem, we have that a homothety H with center S and coefficient k maps where 1).
In order to construct the requested point 1 S 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 1.That figure given in picture 2 consists of a pair of homothetic triangles

C B A
, and which will concurrently be homothetic with triangle , where the center of that homothety is point 0 S , and 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 0 S -the center of the circle inscribed in 0 0 0

C B A
, and let us construct bisectors of its sides in whose intersection is found point 2 S -the center of its circumscribed circle (see figure 2).Since the homothetic figure in picture 2 is constructed in the way that it is similar to the homothetic figure in picture 1, it means that the homothety coefficent is.

Construction (first way):
Through analysis we found that a pair of homothetic triangles ( ) of the figure in picture 2 has the homothety coefficient that equals the one of a pair of corresponding homothetic triangles ( ABC and

S A S
) (see figure 3).
Triangle ABC is given and we can easily construct points S (the intersection of bisectors of its interior angles) and O (the intersection of bisectors of its sides) (see figure 4).Homothety H with center S and coefficient k maps Therefore, in figure 4, point 1 S is to be constructed, that is the segment

First
similarity to the figure in picture 1, i.e. similarity to a corresponding pair of homothetic triangles ABC and homothety coefficient k .We obtain this figure by constructing in the following way: to a pair of homothetic triangles ABC and 1 1 1 Let us construct three congruent circumferences which have one common point 2 S (the center of the circumscribed circle around a part of the construction in figure2, precisely let us construct a pair of corresponding homothetic triangles separately ( SO we easily find the requested point 1 S of triangle .ABC In the similar way we can construct vertexes of the three congruent circumferences in ABC that have the common intersection in point 1 S (figure4).