Problems on maxima and minima arise naturally not only in science and engineering
and their applications but also in daily life. A great variety of these have
geometric nature: finding the shortest path between two objects satisfying certain
conditions or a figure of minimal perimeter, area, or volume is a type of problem
frequently met. Not surprisingly, people have been dealing with such problems for
a very long time. Some of them, now regarded as famous, were dealt with by the
ancient Greeks, whose intuition allowed them to discover the solutions of these
problems even though for many of them they did not have the mathematical tools
to provide rigorous proofs.
For example, one might mention here Heron`s (first century CE) discovery that
the light ray in space incominga point A and outgoing through a point B
after reflection at a mirror α travels the shortest possible pathA to B having
a common point with α.
Another famous problem, the so-called isoperimetric problem, was considered
for example by Descartes (1596-1650): Ofplane figures with a given perimeter,
find the one with greatest area. That the "perfect figure" solving the problem
is the circle was known to Descartes (and possibly much earlier) however, a rigorous
proof that this is indeed the solution was first given by Jacob Steiner in the
A slightly different isoperimetric problem is attributed to Dido, the legendary
queen of Carthage. She was allowed by the natives to purchase a piece of land
on the coast of Africa "not larger than what an oxhide can surround." Cutting the
oxhide into narrow strips, she made a long string with which she was supposed to
surround as large as possible area on the seashore. How to do this in an optimal
way is a problem closely related to the previous one, and in fact a solution is easily
found once one knows the maximizing property of the circle.
Another problem that is both interesting and easy to state was posed in 1775
by I. F. Fagnano: Inscribe a triangle of minimal perimeter in a given acute-angled
triangle. An elegant solution to this relatively simple "network problem" was given
by Hermann Schwarz (1843-1921).
Most of these classical problems are discussed in Chapter 1, which presents
several different methods for solving geometric problems on maxima and minima.
One of these concerns applications of geometric transformations, e.g., reflection
through a line or plane, rotation. The second is about appropriate use of inequalities.
Another analytic method is the application of toolsthe differential calculus.
The last two methods considered in Chapter 1 are more geometric in nature
these are the method of partial variation and the tangency principle. Their names
speak for themselves.
Chapter 2 is devoted to several types of geometric problems on maxima and
minima that are frequently met. Here for example we discuss a variety of isoperimetric
problems similar in nature to the ones mentioned above. Various distinguished
points in the triangle and the tetrahedron can be described as the solutions
of some specific problems on maxima or minima. Section 2.2 considers examples
of this kind. An interesting type of problem, called Malfatti`s problems, are contained
in Section 2.3 these concern the positioning of several disks in a given figure
in the plane so that the sum of the areas of the disks is maximal. Section 2.4 deals
with some problems on maxima and minima arising in combinatorial geometry.
Chapter 3 collects some geometric problems on maxima and minima that could
not be put into any of the first two chapters.
Finally, Chapter 4 provides solutions
and hints toproblems considered in the first three chapters.
Each section in the book is augmented by exercises and more solid problems
for individual work. To make it easier to follow the arguments in the book a large
number of figures is provided.
The present book is partly based on its Bulgarian version Extremal Problems in
Geometry, written by O. Mushkarov and L. Stoyanov and published in 1989 (see
). This new version retains about half of the contents of the old one.
Altogether the book contains hundreds of geometric problems on maxima or
minima. Despite the great variety of problems considered-from very old and
classical ones like the ones mentioned above to problems discussed very recently
in journal articles or used in various mathematics competitions around the world-
the whole exposition of the book is kept at a sufficiently elementary level so that it
can be understood by high-school students.
Aparttrying to be comprehensive in terms of types of problems and techniques
for their solutions, we have also tried to offer various different levels of
difficulty, thus making the book possible to use by people with different interests
in mathematics, different abilities, and of different age groups. We hope we have
achieved this to a reasonable extent.
The book reflects the experience of the authors as university teachers and as
people who have been deeply involved in various mathematics competitions in
different parts of the world for more than 25 years.
The authors hope that the book will appeal to a wide audience of high-school students and mathematics teachers,
graduate students, professional mathematicians, and puzzle enthusiasts. The book
will be particularly useful to students involved in mathematics competitions around
- Informação Adicional
Nome Real Geometric Problems On Maxima And Minima - Titu Andreescu, Oleg Mushkarov e Luchezar Stoyanov Apelido Geometric Problems on Maxima and Minima Categoria Matemática Autor Não Páginas 0 Idioma Não Editora Não Formato Não ISBN Não Demo http://issuu.com/livrariavestseller/docs/0313?mode=window Parceiro Externo Amazon N/A Parceiro Externo Apple N/A Parceiro Externo Kobo N/A Parceiro Externo Wook N/A Preço Assinante N/A
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