Difficult Problems in Geometry
                  Natalia Malaschonok
                Tambov State University
                natasha@math-iu.tstu.ru

The title is a general name for a class of problems that arise in teaching
mathematics.  The main problem both in secondary and high school is an
upbringing of mathematical intuition.  It is not paradoxical that a way here is
an algorithmization of mathematical problems and cycles of problems.

An example is a geometric problem of an arbitrary character.  A correct
formulation presupposes an existence of an object under consideration.  The
best and universal way to create an algorithm of solving is a construction of
the object.  It is a well-known approach of ancient Greek mathematics.  Now it
is especially actual in methodology of teaching owing to modern computer
systems adopted to the needs of mathematics.  One of the best is
"Mathematica".  It is easy in use and suitable to specialists in various
spheres of teaching mathematics.

So the main idea is a reconstruction of an object that is ought to be
calculated mathematically.  The closest to geometry is mechanics, and a great
amount of applications.

In the talk we demonstrate the examples of such algorithmization of some
problems and of the cycle of problems.  We hope that it must be interesting not
only in teaching but in applications of various sort also.