Abstract
This article discusses the advantages of teachinglearning Geometry with the help of the new educational technologies to motivate students and undergraduates, which may lead to building focus, interactivity and feedback during lessons. The 3rdyear students from the Department of Mathematics, “Vasile Alecsandri” University of Bacău, also attending the courses of the Pre and InService Teacher Training Department, have experimented the relevance of the new technologies in teaching and learning Geometry, by applying a new approach to the design and conduct of a lesson of Geometry. The research was conducted during the 20142015 school year, involving two groups, each of them comprising 132 students: 66 children, from 3 classes having 22 students on the average each, experimental group – the 8th A grade from “Octavian Voicu” Middle School, Bacău, and 66 children, from 3 classes having 22 students on the average each, a control group – the 8th B grade from “Miron Costin” Middle School, Bacău and demonstrated the efficiency of teaching Geometry the concurrence of relevant lines in a tetrahedron by using computers in solving problems and promoting Geometry knowledge, building active thinking and competences, as well as skills in representing plane and space geometrical figures.
Keywords: AEL laboratoriesgraphical representationconcurrencetetrahedronexperimental research
Introduction
In order to make the economy of the European Union a most competitive one at the global level, it
is necessary to build a strong foundation for the education of future generations. In order to facilitate the
success of an economy, it is required that the physical persons owning thorough knowledge, key
competences and who are consistent with the latest technologies should firmly apply their expertise and
information to today’s society, as shown in the Lisbon Strategy.
The Lisbon Strategy is a key European orientation that plans to implement a more relevant use of
information and communication strategies in the classroom. The gaps at the educational level are very
serious. The Romanian university education should undertake a European approach in order to create new
and innovating concepts, pedagogical and didactic strategies, to improve teaching and learning(Cerghit,
2002).
The students should build key skills, including digital literacy, foreign languages and – in our
context – Mathematics, through professional training. University training should take into consideration
the students’ teamwork needs within a strong European partnership and focus on improving teaching
learning strategies.
The students’ initial training aims at a fundamental improvement of teachers at a European level.
By introducing the innovative ideas of ICT, present and future teachers should ensure a modern
professional education. This is a prerequisite for building and supporting a highly motivated society, with
a high level of awareness in Europe. (Lupu, 2013).
Building digital educational content is essential for the use of modern information and
communication technologies in mathematical education (both in schools and universities). We shall
present and evaluate the experience in using modern educational software in and outside the classroom.
The teachers from Romanian schools are interested in using AEL laboratories and ICT during
lessons, as well as dynamic mathematical software for teaching and learning. Also, there is required the
use of the interactive board in everyday lessons, hence every school should have an interactive board. The
relevance of the role of teachers as partners of students was highlighted by the use of laboratory
digitalised environments in mathematical training.
A concern of university teachers is to organize courses or workshops on the use of computers in
teaching Mathematics, with the aim of training middle and highschool teachers in this respect.
Mathematical software represents a highly supportive component for encouraging individual learning and
solving Geometrical problems, by providing students and undergraduates with tools that may be
individually used inside, as well as outside the classroom, in the initial training of teachers.
During the Teaching Practice, the students and pupils from the research group used AEL programs
to graphically represent geometrical figures, mainly tetrahedrons, highlighting relevant lines and
concurrence properties.
The experimental research consisted in evaluating some initial, formative and final tests that
highlighted the importance of using computers, as well as acquiring the relevant lines in a tetrahedron,
with their concurrence properties.
Theoretical Aspects. The Concurrence of Relevant Lines in a Tetrahedron
We shall analyse some concurrence problems of the following relevant lines in a tetrahedron: the
medians (the segment connecting a point with the centre of gravity of the opposite side), the bimedians
(the segment uniting the middle points of two opposite edges), the perpendiculars to the sides of the
tetrahedron on their centres. There rose the following problemsituations: Are the heights in a tetrahedron
concurrent? Is there a condition if and only if, that ensures that the heights in a tetrahedron are
concurrent? (Lupu, & Săvulescu, 1999, p.84) Problem 1. Let M be the middle of the edge CD of a tetrahedron ABCD, and I∈BM and J∈AM
the points of the centres of gravity of the sides BCD and ACD. Prove that the medians AI and BJ are
concurrent. (Lupu, & Săvulescu, 2000, p. 127)
Demonstration: Because I and J represent the centre’s of gravity of sides BCD and ACD, they will
!!
be located on medians BM and AM at ! from CD and ! from vertex B and A. In triangle ABM, ∥ 𝐴𝐵
!"!"!"!"!"!
and let there be 𝐴𝐼 ∩ 𝐵𝐽 = 𝐺 , then !"=!"=!"=!"=!"=! . The medians of the tetrahedron,
being coplanar two by two, are also concurrent two by two, therefore they have a common G point that
divides each median according to the relation1/3 and is called the centre of gravity of the tetrahedron.
Regarding the concurrence of medians, there is demonstrated that: In a tetrahedron ABCD, the medians MP, NQ are RS are concurrent. (M∈BC, MB≡MC ; N∈CD , NC≡ND ; P∈AD , PA≡PD ; Q∈AB , QA≡QB
; R∈AC, RA≡RC ; S∈BD , SA≡SO).
For demonstration, observe that the quadrilateral MNPQ is a parallelogram, and QN and MP are
concurrent, being diagonals in this parallelogram and cut into half. By analogy, RS cuts the middle of QN
and MP. We may further demonstrate that the concurrence point of the medians is also the centre of
gravity G, because the median MP, for example, is the median of the median plane (MAD) and of the
median plane (NBD), and since the gravity centre belongs to each of the median planes of the
tetrahedron, it results that G also belongs to each median. (Peligrad, Ţurcanu, & Popa , 2015).
Regarding the concurrence of the perpendiculars on the centres of the circles circumscribed to the
lateral sides, we will first solve the fact that the bisecting planes of the sides of a triangle have a common
line. Building the bisecting planes and of sides AB and BC from ABC we will have ∩ = PO and 𝑃𝐴 ≡ 𝑃𝐵 ≡ 𝑃𝐶. Line PO of the intersection of the bisecting planes of triangle ABC is the median in space of this triangle and every point on it is equally distanced from the triangle tips.
Then, we demonstrate that the six bisecting planes of the sides of a tetrahedron are concurrent in a point that is equally distanced from the vertices of the tetrahedron. If m1 is the median in space of the plane of triangle ABC and the bisecting plane of the edge AD where O = m1∩, we have: 𝑂𝐴 ≡ 𝑂𝐵 ≡
𝑂𝐶 ≡ 𝑂𝐷 = 𝑅 which shows that O also belongs to the other bisecting planes.
The sphere with the centre O and radius R is called the sphere circumscribed to the tetrahedron.
Are the heights concurrent? The answer is no. We may illustrate it with an example: in the tetrahedron
DABC with the basis an equilateral ABC and DA ⊥ (ABC), the heights BM, CN and DA are not
concurrent. Regarding the fact that the heights of a tetrahedron should be concurrent, we shall firstly
define the orthocentric or orthogonal tetrahedron (a term coined by Steiner in 1927), namely, a
tetrahedron is orthocentric if its opposing edges are perpendicular.
Lemma: If in the tetrahedron ABCD two pairs of opposing edges are perpendicular, then the other
two edges are also perpendicular.
Demonstration: Let there be AB ⊥ CD and BC ⊥ AD. Draw AE ⊥ DC and AF ⊥ BC. It results CD ⊥ (ABE) and BC ⊥ (ADF). If AAʹ = (ABE)∩ (ADF) then AAʹ⊥ (BCD). But CAʹ⊥BD and therefore
BD⊥(ACAʹ), it results BD ⊥ AC. Then, demonstrate the theorems.
Theorem 1: The heights of a tetrahedron are concurrent if and only if the tetrahedron is orthocentric.
Demonstration: We demonstrate that the heights from A and D are concurrent when AD⊥BC. Let
Aʹ and Dʹ be the orthogonal projections of points A and D on planes (DBC) and (ABC) and AAʹ and DDʹ
concurrent, AAʹ⊥(DBC)⇒AAʹ⊥BC and DDʹ⊥(ABC)⇒DDʹ⊥BC. Therefore, BC being perpendicular to
plane (AAʹ,DDʹ), it is therefore also perpendicular to AD. Reciprocally, if AD⊥BC, let F be the foot of
the perpendicular from A to BC. From BC⊥AD and BC⊥AF it results that BC⊥(AFD)⇒BC⊥AAʹ .
Analogously, BC⊥DDʹ. Since AAʹ⊂(AFD) and DDʹ⊂(AFD) it results that AAʹ and DDʹ are heights in
triangle (AFD), therefore are concurrent. Hence, it results that if the heights AAʹ and DDʹ are concurrent,
then the other two heights BBʹ and CCʹ are also concurrent. Indeed, AAʹ∩DDʹ∅⇒ AD ⊥ BC therefore
BBʹ∩ CCʹ∅.(Lupu, 2014, p. 135).
We may demonstrate that if heights AAʹ and BBʹ are concurrent in H, then the common
perpendicular of edges AD and BC contains the point H, because if the plane (AAʹ,DDʹ) cuts BC in F, it
results that FH is the third height, therefore FH⊥AD; since BC⊥(AFD) ⇒ FH⊥BC. By analogy, we may
demonstrate that if the heights AAʹ and DDʹ are cut in point H and BBʹ and CCʹ in Hʹ≠ H then HHʹ is the
common perpendicular of the opposing edges AD and BC.
From the three conditions of perpendicularity of the orthocentric tetrahedron, it results the
concurrence of heights. Another particularly interesting theorem for analysis is: Theorem 2: A tetrahedron is orthocentric if and only if: 𝐴𝐵!+ 𝐶𝐷!= 𝐴𝐶!+ 𝐵𝐷!= 𝐴𝐷!+ 𝐵𝐶!.
Demonstration: We demonstrate, first of all, that the heights from A and D of tetrahedron ABCD are secant if and only if𝐴𝐵!+ 𝐶𝐷!= 𝐴𝐶!+ 𝐵𝐷!. Let there be F and F1 the orthogonal projections of
points A and D to BC. There occur the relations: 𝐴𝐵!− 𝐴𝐶!= (𝐴𝐹!+ 𝐵𝐹!) − (𝐴𝐹!+ 𝐹𝐶!) =
𝐵𝐹!− 𝐹𝐶!= 𝐵𝐹 + 𝐹𝐶 ∙ 𝐵𝐹 − 𝐹𝐶 = 𝐵𝐶 𝐵𝐹 − 𝐹𝐶 1 ; 𝐵𝐷!− 𝐶𝐷!= (𝐷𝐹!! + 𝐵𝐹!!) −
(𝐷𝐹!+ 𝐹!𝐶!) = 𝐵𝐹!! − 𝐶𝐹!! = 𝐵𝐹! + 𝐶𝐹! ∙ 𝐵𝐹! − 𝐶𝐹! = 𝐵𝐶 𝐵𝐹! − 𝐶𝐹! 2 .
It can be observed that AAʹ and DDʹ are secant if and only if F coincides with 𝐹!. From the
previous equalities with 𝐹 = 𝐹! ⇒𝐴𝐵!− 𝐴𝐶!= 𝐵𝐷!− 𝐶𝐷!⇒𝐴𝐵!+ 𝐶𝐷!= 𝐵𝐷!+ 𝐴𝐶!.
Reciprocally 𝐴𝐵!+ 𝐶𝐷!= 𝐵𝐷!+ 𝐴𝐶!⇒ 𝐵𝐶 𝐵𝐹 − 𝐹𝐶 = 𝐵𝐶(𝐵𝐹! − 𝐹!) therefore 𝐵𝐹 − 𝐹𝐶 =
𝐵𝐹 + 𝐹𝐹! −
− 𝐹𝐹! + 𝐹𝐶 = 2𝐹𝐹! = 0 therefore F and 𝐹! coincide. Demonstrating analogously and taking
into consideration the concurrence of all the other heights, it results: 𝐴𝐵!+ 𝐶𝐷!= 𝐴𝐶!+ 𝐵𝐷!= 𝐴𝐷!+
𝐵𝐶!. We can also easily deduce the fact that the tetrahedron ABCD is orthocentric if and only if its three
bimedians are congruent(Lupu, 2015, p. 850).
Research Description
Research Objectives
By using the proper strategies in teaching Mathematics, we may achieve a higher learning
performance at 8thgraders. The undergraduates were preoccupied with equipping students with thorough
skills and abilities in using mathematical software, thus optimizing the process of learning Mathematics.
In this respect, there were established several objectives for guiding and orienting the whole
investigation:  theoretical classification of the contexts approached from the perspective of the theme; 
initial evaluation of the level of mathematical knowledge, skills and abilities of students, at the beginning
of the 1st semester, in the 20142015 school year, at two 8thgrade classes;  evaluation of computer skills;
 final evaluation of the level of knowledge following the application of the progress factor, respectively
the concurrence of relevant lines in a tetrahedron;  formulating the conclusions of the study (Landsheere
1979).
Research Hypothesis
To achieve these objectives and verify the role of solving and composing problems in building the
creative potential of small students, we have formulated the following hypothesis: if we use the new
technologies in teachinglearning Geometry in the concurrence of relevant lines in a tetrahedron, then we
shall contribute to building intellectual skills and the creative potential of the middleschool student, as
well as enhance the concurrence of relevant lines in a tetrahedron.
Research Methods
In our research, we have used the following methods: observation; conversation; evaluation test;
analysis of activity products. We have used the formative type of psychopedagogical experiment in order
to verify the impact of using activeparticipative methods and modern means and techniques for teaching
learningevaluating elements of Geometry related to the methods of demonstrating concurrence lines in a
tetrahedron. The statistical method was used to process, analyze, systematize and interpret the data
obtained following the measurements. The data from the synthetic tables were graphically represented
through: histograms, frequency polygons and round diagrams (Cristea, 2003).
Research Stages and Sample
The research was conducted during the 20142015 school year, involving two groups, each of
them comprising 132 students: 66 children, from 3 classes having 22 students on the average each,
experimental group – the 8th A grade from “Octavian Voicu” Middle School, Bacău, and 66 children,
from 3 classes having 22 students on the average each, a control group – the 8th B grade from “Miron
Costin” Middle School, Bacău.
The basic method used was the pedagogical, observationalformative experiment, applied through
the method of the tests. The research involved several stages:
1. The observational stage that covered the interval October 15th – October 30th. At this stage,
there were applied tests for identifying the initial level of the students’ digital and intellectual skills.
2. The ameliorative stage, conducted over the 1st and 2nd semester, November 1st – May 28th. At
this stage, there were organized differentiated training activities, aimed at improving digital competences
by using mathematical software (progress factors within the conducted experiment). The analysis of the
results from the evaluation tests enabled the implementation of differentiated, ameliorative pedagogical
measures. Besides the activities conducted for this purpose, at the lessons from the optional discipline
Educational software, during the sequences of independent activities, the students received individual or
group tasks. Following these activities, the students made visible progress.
3. The stage of final evaluation covered the interval June 1st – June 10th and was aimed at
retesting the digital skills of students, as well as their potential for using the computer in the graphical
representation at the end of the pedagogical experiment.
In our ameliorative experimental research, obtaining evaluation information based on tests with
pedagogical objectives requires coverage of the following stages:  applying the test and obtaining
students’ answers, which are evaluated;  comparing each answer with the standard solution and giving
the corresponding score for correct solutions;  giving school marks by applying the criteria of converting
scores into marks;  interpreting the results and adopting ameliorative measures (Dewey, 1992).
Research Results
This study is a continuation of our previous articles (Lupu, 2014 & Lupu, 2015) where we have
demonstrated that informational technologies hold a privileged position within the set of the factors
responsible for the students’ school success and how using computers and active, cooperation methods during
Geometry lessons has a positive effect upon students, supporting the building of digital, communication and
teamwork skills.
The analysis of the analytical and synthetic tables, the histogram, the frequency polygon and the
circular diagram generated the following conclusions regarding the initial evaluation: the experimental
group has achieved the following results: of the 66 evaluated children, 29 obtained the mark of very well,
representing 44% of them, 24 children obtained the mark of well, meaning 36%, and 13 children obtained
the mark sufficient, representing 20% of the participants; the control group achieved the following
results: of the 66 evaluated children, 29 obtained the mark very well, representing 44% of them, 26
children obtained the mark well, namely 40%, and 8 children obtained the mark sufficient, representing
12% of the participants, whereas 3 children – 4% got insufficient.
The application of the initial test enabled the identification of the students’ gaps and the extent of
these gaps, the prolonged emphasis on the concurrence of relevant lines until all the students have
reached a corresponding training level. In Table 1 and frequency polygon 1, shows the comparative
analysis of the initial evaluation for the two groups.
At the current, formative stage, in the experimental class the focus was particularly on issues
related to the use of computers in the graphical representation of planes and geometrical figures, on
solving and demonstrating problems.
Thus, there were regularly applied, during lessons of Mathematics, formative evaluation tests.
Thus, the students solved tests based on the hypothesis that the permanentization of immediate control, as
an extrinsic motivational situation, would build the motivation needed in building digital skills,
imagination and creativity by solving problems through the use of the computer in graphical
representation.
The formative evaluation tests applied in lessons of Geometry enabled the immediate
identification of the students’ learning difficulties. In order to eliminate mistakes, the activity was
differentiated. Following the analysis of tests, there were presented the operational objectives that were
not achieved by students, so that they may be aimed at in the proposed recovery activities.
The analysis of the synthetic table no. 2, the histogram and the frequency polygon reveals the fact
that, in the final evaluation:  for the experimental group, the results were the following: of the 66
evaluated children, 37 obtained the mark very well, representing 56% of them, an increase by 12% , 26
children obtained the mark well, their percentage also being 40 %, and 3 child got the mark S (sufficient),
representing 4% of the participants;  for the control group, the results were the following: of the 66
evaluated children, 29 obtained the mark very well, representing 44% of them, 26 children obtained the
mark well, their percentage also being 40%, 8 children got the mark sufficient, representing 12 % of the
participants, and 3 child insufficient, representing 4 % .
Calculating the average between the two tests (initial and final), and drawing a comparison
between the two groups, there may be observed an increase in the school performance at the experimental
group, compared with the control group.
The comparative analysis of the histogram and frequency polygon no. 2 reveals the progress
recorded by the experimental group at the end of the experiment. Calculating the average between the
initial and final evaluation, there were obtained the following results.
Conclusions
At the theoretical level, there were created the premises for conceptualizing the teaching strategies
for solving problems of Geometry in the concurrence of relevant lines in a tetrahedron. The analysis of
the experimental test from the two classes supports the formulation the following conclusions:  the
informational technologies hold a privileged position within the set of the factors responsible for the
students’ school success;  using computers and active, cooperation methods during Geometry lessons has
a positive effect upon students, supporting the building of digital, communication and teamwork skills; 
the efficient use of AEL laboratories and the use of software characteristic of geometrical representation,
as well as activeparticipative methods constitutes a challenge, both for students and the teacher.
Following the application of the ameliorative experiment by using the method of the tests, as well as of
active, cooperation methods in teaching, it was found that their use with a specific purpose, at the right
moment, may generate satisfying results, such as:  the students overcame their communication
blockages;  they built skills in solving Geometry problems;  they manifested proper behaviour towards
their colleagues and group;  there was better cooperation among children, these became more tolerant; 
there was a combination of types of work (frontal, group, individual), which created great possibilities for
the multiple and diverse activation of students;  the students learnt that in order to achieve a group task,
they need one another.
The results obtained by the students validated the research hypothesis. The use of modern
technologies in the activity of solving Geometry problems contributes to optimizing learning and
rendering it efficient, stimulating the students’ intellectual and creative potential, obtaining performances
according to age and individual particularities.
In conclusion, we may say that in order to achieve quality education and obtain the best results, we
should apply the project “InnoMathEd – Innovations in Mathematics” and combine traditional and
modern methods in the teachinglearningevaluation of Mathematics.
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Publication Date
25 May 2017
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Educational strategies, educational policy, organization of education, management of education, teacher, teacher training
Cite this article as:
Lupu, C. (2017). Methods Of Demonstrating Concurrence Lines In A Tetrahedron. In E. Soare, & C. Langa (Eds.), Education Facing Contemporary World Issues, vol 23. European Proceedings of Social and Behavioural Sciences (pp. 13631371). Future Academy. https://doi.org/10.15405/epsbs.2017.05.02.167