THE METHOD OF MIDDLE VALUES ABOUT PROVING
INEQUALITIES BETWEEN ELEMENTS OF THE TRIANGLE
Zhaneta Germanova Germanova
Galabovo, BULGARIA
Abstract. The article presented three systems with tasks about learning
the method of middle values to proving inequalities between elements of the
triangle. The solving the problems in the systems is based on the well-known
inequalities between middle values (the inequality of Koshi, the inequality
between average harmonic and average, the inequality between average and
mean square and inequality between the average degrees and extent of the
average).
The proposed systems of tasks complements, extends and systematizes
the knowledge of trainees about triangle relationships.
The article is intended for teachers working with gifted students in
mathematics in the 11th and 12th grades and preparing students for application
in higher education
In the article we present three systems of tasks to master the method of the
middle values by proving the triangle inequalities.
The tasks within each system № № 2-4 are arranged in order of increasing
complexity, as we had in mind the connections between them, i.e. previous task
can be used as task-components by proving the following tasks on the systems.
Moreover, at the selection and compilation of the tasks included and such that
can be solved in different ways, i.e. by using various knowledge acquired by
students during their training.
The selection of tasks on the compilation of the systems is based on the
analysis of school curricula and textbooks, firstly, the possibilities for
application of system tasks № 1 (as a supporting) in the following systems I (№
№ 2-4 ), on the other, and based on formulated by K. Garov in [2, p.16] general
requirements to systems of tasks in the school informatics.
Well-known inequalities between the averages (Cauchy inequality,
inequality between the average harmonic and the mean average, etc.) are used as
knowledge-method for proving triangle inequalities. Their use with a general
method gives more rational ways of solving mathematical problems.
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The training of gifted students from 11th and 12th graders in
extracurricular activity in mathematics can be realized in several exercises over
the following technologies:
1. Preconditioning for: introduction to theoretical bases of applied
knowledge (inequalities between the averages).
2. Supporting apparatus needed as a tool to prove the triangle inequality.
3. Application of the inequalities between the averages for proving the
triangle inequalities.
Supporting apparatus needed as a tool to prove the triangle inequality:
SYSTEM № 1
BASIC EQUATIONS AND INEQUALITIES IN THE
SCHOOL COURSE IN MATHEMATICS
This system aims to update the basic equations and inequalities from
school mathematics course by which to optimize the formation of skills to prove
triangle inequalities. These knowledge are the basis for the implementation of
other systems.
1. Basic algebraic inequalities:
а)
(property)
b)
,
,
(options for presentation
the inequality of point а))
c)
2. Inequalities between the triangle elements:
a)
,
,
b)
,
,
c)
,
,
d)
,
,
3. Formulas for the triangle area:
а)
, where a, b, c are lengths of the triangle sides, and
, – of their respective heights;
b)
, where p is a half the triangle perimeter, and r is the radius of a
circle inscribed in a triangle;
c)
, where , and are radii on
outsides inscribed circles for a triangle;
d)
, where is the radius of the circle described about the triangle;
,
e)
angles;
f)
g)
, where
,
,
are measures of the triangle
(Heron's formula);
.
2
4. Formulas, expressing the some elements of the triangle by the its sides,
the surface (B) or half the triangle perimeter (p):
a)
,
,
(followed by 3a));
b)
,
, where
c)
and
,
are the triangle medians;
,
, where ,
,
and
d)
, (followed by 3 d));
e)
(followed by 3 b));
f)
,
,
,
are the bisectors of the triangle;
(followed by 3 c)).
Application of the inequalities between the averages for proving
inequalities between elements of the triangle.
SYSTEM № 2
SYSTEM OF TASKS FOR USING OF SYNTHETIC
APPLICATION OF INEQUALITY CAUCHI
METHOD
THROUGH
With a view to developing skills for application of the synthetic method in
solving problems, here we use the relationship between the average and the
geometric mean of the given positive numbers. The tasks of the system, relate to
inequalities between triangle elements.
1. Be prove the inequality
To proving the inequality is appropriate to use a synthetic method. For
numbers p-a>0, p-b>0 and p-c>0 to apply Cauchy inequality and formulas of
the triangle (from system 1, 3b) and 3f)).
2. Be prove the inequalities:
а)
,
,
To proving the inequalities is appropriate to apply the Cauchy inequality
for positive numbers р–a, р–b, р–c.
b) R ≥ 2.r (Euler's inequality).
Given formulas 3d) and 3b) of a system, is appropriate to be multiplying
the three inequalities of the previous paragraph a).
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(Euler's inequality there are applications to prove of other inequalities).
3. Be prove the inequalities:
а)
,
,
b)
c)
To proof of inequalities in task 3a) it is required from teacher to using
with students the updated formulas 4c) of system 1 and the inequality between
the average and the geometric mean of two positive numbers.
In the proof of inequality in task 3b) using task 3a).
In the proof of inequality in task 3c) is appropriate to consider the obvious
inequalities, with a decision of this task is to task 3b).
4. Be prove the inequality
To prove the inequality is appropriate to apply the respective formulas 4a)
and 4e) of system 1 and the Cauchy inequality.
5. Be prove the inequality
In the proof of inequality through the synthetic method is appropriate to
use the inequality between the arithmetic mean and the geometric mean and task
4.
6. Be prove the inequality
To prove this inequality in the synthetic method is necessary to use
formulas 2c), 4a) and 3b) of system 1 and the Cauchy inequality.
7. Be prove the inequality
To prove the inequality can be used the idea of task 5.
8. Be prove the inequality
To proving at the task is appropriate to use the synthetic method by
applying Cauchy inequality for each pair of multipliers on the left side of the
given inequality and to multiplying on the obtained results.
9. Be prove the inequality
4
In this task can be applied, except Cauchy inequality, and also formulas
4a) and 4d) of system 1.
A main aim on the considered system of tasks is the formation of skills of
learners to use the Cauchy inequality with realization of the synthetic method.
An overview of system tasks № 2 and prevailing methodological notes there
give grounds to conclude that in the evidences of most of the discussed
inequalities must to used synthetic reasoning. Characteristic of this system is
that there is a strong link between successive tasks in it (the idea is implemented
for basic tasks and task-components), and between the methods used to solve
them.
A look on the structure of the inequalities of a system 2 (see tasks 2a, 4, 5)
shows that : in the presence of sum of three addends or work of three multipliers
(in one the sides of the given inequality) existed a relationship between the mean
average and the geometric mean and can we are apply Cauchy inequality as a
means of solving the above problems. Given these conditions, can say that is
easily to realize the synthetic method, as a departure point is the support task - in
this case the Cauchy inequality.
In the event that the application of the Cauchy inequality (such as a
support task) is not obvious, then it is appropriate to seek and discover other key
relationships arising from the types of inequality (i.e. the condition of the task)
and to derive new information (essential for the proof).
Namely, here it is necessary, to come insight, insight as a result of greater
observation of the learner and good reflective skills. Knowledge of analytical
methods for proving inequalities in a triangle directs a learner for apply
knowledge from school mathematics course (see system № 1): identities,
formulas, relationships that will support evidence sought in the quality of
support tasks. As an example, proving the inequality of problem 3a requires that
you firstly apply formula 4c) of the system № 1, and then - the Cauchy
inequality.
SYSTEM № 3
SYSTEM OF TASKS ABOUT MASTERING OF THE RELATIONSHIP BETWEEN
HARMONIC MEAN AND AVERAGE, AVERAGE AND ROOT MEAN SQUARED (
) , т.е.
(
)
For use of these connections from the students we offer the following set
of tasks.
1. Be prove the inequality
, where a, b and c are positive numbers, i.e.
inequality is particularly true for the sides of each triangle.
5
The proof of the inequality follows directly from the specific application
of the inequality between the harmonic mean and the arithmetic mean (
)
for numbers a, b and c. The example and its solution serves as a model for the
students to use the synthetic method for its implementation using the harmonic
mean inequality and the arithmetic mean for proving inequalities between
elements of the triangle.
2. Be prove that for the sides a, b, c and half the triangle perimeter p in an
arbitrary triangle is true the inequality
.
To proving the inequality in this task are using arguments similar to those
of task 1. Thus, students assimilated the ideas set out in the previous task.
3. Be prove that for the sides a, b and c and for a half triangle perimeter p in
an arbitrary triangle is true inequality
The proof of this inequality illustrate the application of the inequality
between the mean average and the mean square (
) – in the case of the
three numbers a, b and c. We can say that this task is also a model for the
students to using the synthetic method using the above inequality.
4. Be prove the inequality that is valid for any triangle
By solving a task can be reinforced the student's knowledge for
application of the inequality between the mean average and the mean square
(
) - in the case of three numbers , , and . Furthermore, from system
№ 2 is used here task 7, which plays role of a task-component.
5. For each triangle is in force
And in this task, the primary method for solving is the synthetic, like in
the process of its implementation be used the inequality between the mean
average and the mean square for the three positive integers
,
and
.
And in constructing this system we have strived to the selected tasks are
arranged on the principle of mutual relation between them and the used ideas
and the method for solving the above problems can be applying for next .
As a result of our collaboration with students from grade 11 and 12 on the
topic of system 3, we reach the following conclusions:
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1. Giving the individual work of students home creates interest in
advance and activity in the coming exercices.
2. The presence on a sum from three addend in an inequality (as in cases
where tasks 1, 2) from the type
reminds us that we can look a relation
between the average harmonic and the mean average, i.e. we are using the
remarkable inequality (for n = 3)
as a means of solving
the tasks.
3. Remarkably that if one side of an inequality contains the sum of the
squares of three addend from the type a2+b2+c2 (see tasks 3, 4, 5), in this case is
appropriate to use the relation between the mean average and the square mean ,
i.e.
when n = 3.
SYSTEM № 4
SYSTEM OF TASKS RELATED WITH THE USE OF INEQUALITY
,
1. Be prove that
,
,
To proving the inequality
median
can be used the formula for the
and inequality
.
Similarly proceed at consideration of the next inequalities in the task.
2. Be prove the inequality
To proving this inequality be used different ideas.
One of them is the use of inequality between the mean average and the mean
square (see task 5 of the system № 3).
Another idea is associated with use of the inequality
, where for k = 2 is obtained
.
3. Be prove the inequality
To proving the given inequality play a basic role the inequality
for k = 2 and n = 3 in combination with task 5 of
System 2 as a task-component.
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4. Be prove the inequality
In the proof of inequality can be apply the ideas indicated in task 3.
5. Be prove the inequality
As on the proof of task 3 and here plays a key role the inequality
for к=2 and n=3 in combination with task 9 of
System 2 as a task - component.
In the construction of this system of tasks, except mutual relation between
them, is presence and on the use of tasks by previous systems (for example the
system № 2, № 3, etc.).
In the held exercise on the topic of system 4 are improving the skills of
the trainees to use the synthetic method.
Here they are introduced to yet another remarkable inequality
(
, which apply as
a support task to be proving inequalities.
1. It is appropriate to note that in the inequalities (see tasks 2, 3, 4, 5)
there is the sum of the squares of the three types of addend a2 + b2 + c2 (as seen
in the exerciser of the system 3), we can use dependency expressed in the above
inequality for n = 3.
2. Remarkably that the tasks of this exercise can be proved with use of
the relationship between the averages and the mean square as the "second
means" to solving them.
By using different ideas for proving inequalities students learn to think
and reflect.
As a result of our training activity can to do following conclusions:
1. The development of didactic systems of tasks in different topics can
contribute to successful preparation of students on concerned subjects.
2. The purpose of a constructed system of tasks is students acquire a
certain amount of knowledge in the form of support problems (the school course
and some remarkable inequalities) and to master specific methods (general or
private) to be prove the triangle inequalities for to be able to successfully attack
the proposed inequalities.
3. Purpose on the supporting tasks is to assist the opening of the
decision. In systems № №2-4 easily realize the synthetic method as departure
point is the support task Cauchy inequality, inequality between the averages and
more.
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4. Through a systems of tasks students strive to be directed towards
autonomy in the learning process (to self and self).
5. The relationship between the tasks in the systems allows for conscious
reflection on the utilization of the triangle inequalities.
6. On the selections of inequalities in the systems we had in mind that
"helps algebra to geometry" by application of remarkable inequalities: Cauchy
inequality, inequalities between the averages, as on the support tasks of the
school course in algebra
Educating students in 11th and 12th grades in extracurricular activities in
the proof of the triangle inequalities through the application of inequalities
between the averages in combination with synthetic method increases their
mathematical training, giving them new ideas for solutions, training flexibility
of their thinking that would be useful for rapid adaptation in dynamically
changing life situations.
References
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елементи на триъгълника. /монография/,Пловдив, Изд. “Бойкинг“, 2005.
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ученици за участие в олимпиади и състезания по информатика и
информационни технологии, Автореферат, С.,2008.
3. Милушев, В., Д. Френкев. Дейности, свързани със съставяне на
дидактически системи задачи от определени видове. – Научни трудове на
ПУ “Паисий Хилендарски” т. 42, кн. 2, 2005, 49-60
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