Triangle

This paper presents the role of tools provided by the well known educational software Cabri-Geometry II  [1] or the construction of the concept of similarity in triangles by the students. An a priori analysis showed that a variety of solution strategies could be invented by the students to tackle this concept in the context of the tools of Cabri-Geometry II. In fact, at least seventeen solution strategies can be performed by the students. For the construction of these strategies a variety of geometrical concepts could be integrated with the concept of similarity in triangles such as: the specific circles of a triangle (circumscribed circle, inscribed circle, and escribed circle), regular polygons, different forms of triangles (isosceles triangles, right-angled triangles), specific linear elements of a triangle (altitude of a triangle, bisector of an angle; medians of a triangle, parallel segments joining the mid points of two sides of a triangle), parallel and perpendicular lines, angles,  as well as specific elements of a circle such as tangents of a circle and chords intersected outside and inside of it.

Remember the How Many Triangles game?
How many hours do we spend to find the right answer. but still after all these hours, mathematically speaking the answer is not correct , and this is the proof to the formula to find the number of the triangles drawn.

Triangle has always been the object of study, it also plays a lot of roles in people's daily life. We can learn some theorems, formulas and so on about triangles from our teaching books. These contents can not help us to think deeply. Are these principles true?

So, The purpose of the report is to expplore the principles of triangle by pracatical experiment. I will use some true applications to mimik the triangle and calculate the strenth some neccessary factors. Such as the area, volume.

Any cyclic quadrilateral whose sides are not parallel can define a triangle with one vertex at the point of intersection of the quadrilateral’s diagonals and the other vertices at the points of intersection of the continuations of the quadrilateral’s pairs of opposite sides. Using a cyclic quadrilateral and this triangle, the following four circles may be defined: the circumcircle of the quadrilateral, two circles with diameters that are the sides of the triangle that issue from the point of intersection of the quadrilaterals diagonals, and the Euler circle of the triangle. In the current paper, we shall prove four properties that relate the quadrilateral, the triangle and these circles. We shall show that the two circles whose diameters are the sides of the triangle are perpendicular to the circumcircle of the quadrilateral. We shall prove equalities that relate the angle between the midlines in the quadrilateral with other angles. We shall show that the point of intersection of th...

CV. Dikasoftindo adalah perusahaan Internet Service Provider, yang tidak hanya mengerjakan Sistem Jaringan Internet, tapi juga mengerjakan beberapa proyek Sistem Informasi, Pemrograman, Design Web, Pemeliharaan Lab, Pengadaan Komputer dan Sistem Sekuriti.

RUSUMEN En este artículo se presenta una demostración sencilla de que no existen más que dos triángulos rectángulos de lados enteros en la que el área del triángulo es igual a su perímetro.

Palabras claves: Ternas pitagóricas y Triangulo rectángulo.

In many programs of study, the material on the formulas relating the sides and special segments in a triangle does not appear as part of the study of mathematics in high school. On the other hand, in many programs of study the background required to understand this subject is studied already by the ages of 13-15. This situation gives us the opportunity to teach the relationship formulas at an early stage, even before the studies of geometry have begun in the precise manner at the higher level of difficulty. In this paper we propose a structure and a method for teaching the relationship formulas that has been tried with a group of students. Teaching the relationship formulas by this manner will present the students with many uses for material that has already been studied, and will expose them to new methods for solving problems in geometry and algebra. This paper presents material that is suitable for students aged 13-15. The material includes: (1) Obtaining three formulas that relate special segments in the triangle to the sides of the triangle; (2) Using these formulas for proving three geometrical theorems; (3) Examples of problems in geometry that can be solved algebraically using these formulas; (4) Didactic recommendations for teaching this material.

Precedendo di qualche giorno la presentazione di Context, a FITS SibFest in Romania, (https://www.sibfest.ro/ro/events/context-solo) ho pensato di presentarvi la mia intervista ad Alessandro Carboni. L'intervista risale all'estate scorsa quando, durante l'edizione del Festival Kilowatt "Questa Fervida Pazienza" svoltasi a Sansepolcro, Alessandro Carboni ha presentato, in prima nazionale, le tre creazioni performative: CONTEXT/trio, CONTEXT/duo, CONTEXT/solo e per la prima volta anche CONTEXT / visual, una mostra ad ingresso gratuito, localizzata, per quella occasione, nell'intimo Oratorio di Santa Maria delle Grazie a Sansepolcro. Nella mostra sono stati esposti i disegni ed i materiali preparatori, di studio, ricerca ed approfondimento utili a comprendere meglio gli spazi di interesse di CONTEXT: disegni dei corpi in movimento, misurazioni, mappe territoriali, pezzi di minerali e roccia. Materiali che hanno avvicinato i visitatori ed il pubblico alla complessità dell'intero processo di ricerca presentato dall'artista. Tutte e tre le performance presentate, fruibili anche separatamente, e gli oggetti esposti nella mostra, incarnano da punti di vista diversi l'intero processo di ricerca. Segue l'interessante dialogo che ho avuto il piacere di scambiare con l'artista. Claudia Roselli: Alessandro grazie per aver accettato di fare questa intervista. Puoi presentarti come Alessandro Carboni: performer, artista, artista visuale, ricercatore? Alessandro Carboni: Negli anni ho cercato di ridurre alla parola artista la definizione di quello che faccio. Perché, coltivando vari ambiti, in vari interessi ed in varie discipline i progetti che ho portato avanti hanno veramente attraversato l'arte contemporanea. E' emersa l'esigenza di racchiudere in una sola parola, ovvero "artista visivo", quello che faccio, anche perché più ampia in possibilità. Perché pur attraversando il corpo, la performance, la cartografia, oppure l'architettura, il corpo, eccetera, io ho sempre comunque lavorato a partire dall'idea visiva: anche nel movimento, anche nella coreografia. Quindi questa è la motivazione per la quale preferisco utilizzare la parola artista visivo. Mi pare la definizione più giusta, perchè il filo che ha unito tutte queste pratiche negli anni è stata questa forte relazione con il disegno. Il disegno come idea, come transfer visivo di un pensiero, ma anche disegno come azione, disegno come traccia, disegno come cartografia anche dell'anima.

El presente artículo de divulgación científica, ofrece una demostración de la fórmula clásica para calcular el área de un triángulo (base por altura sobre dos), para ello,  empleará dos de las herramientas más importantes de la Matemática: La Geometría Analítica y El Cálculo Integral.

In Geometric, Triangle is one of the two-dimensional plane having three sides and which is very important to solve problems related to Geometry and sometimes in other subject also. Some fundamental concept /theorems of triangles are required to solve such problems and such theorems are (i) Pythagoras 1st theorem[1] (ii) Pythagoras 2nd theorem (iii) Appollonius theorem (iv) Euclid’s 1st theorem[2] and (v) Euclid’s 2nd theorem (Altitude theorem) [3]. In addition to these, Author has attempted to develop a new formula related to triangle. By this theorem the value of ‘d’ in any scalene triangle can be easily evaluated. While doing calculation related to Geometry, sometimes it is required to additional formulae and it needs extra manpower time and complete basic knowledge about fundamental concept and theorems. The author has now derived some new formulas from the basic mathematical concepts and theorems are as follows:
1.  Formulae for Scalene Triangle (4 Nos.)
2.  Formulae for Right-angled Triangle (2 Nos.)
3.  Formulae for Parallelogram (1 No.)
These formulae are shown in the mathematical expressions also and these may be used to determine any one of the variables without using any trigonometric functions or formulae such as (i) Sine rule, (ii) Cosine rule, (iii) Pythagoras theorem, (iv) Projection rule and etc., These are in readymade form and these are very useful for those working or engaged with the related subject.

Communication au 43 e Congrès de la Société de Mythologie Française à Eymouthiers (Charente) le 26 aout 2021. Culte des baleines en Morbihan au néolithique ancien et Chronicques Gargantuines. Labilité et continuité des récits mythologiques. Intérêt de la notion de déités « doubles ». Jacques Berruchon. Résumé. Le double est défini comme un Esprit similaire à un autre Esprit. Indépendants, ils ont une origine commune et/ou une fonction commune. Le champ d'investigation est la mythologie bretonne, française voire européenne. Nous choisissons un point de départ le plus reculé possible dans le temps. La mise en regard des gravures rupestres des environs de Locmariaquer et des Chroniques gargantuines livre de nombreuses similitudes qui permettent de définir trois « rangs » divins présents depuis la fin du mésolithique. En premier les grands dieux, doubles abstraits qui sont : une grande déesse incarnée par un agrenon à maille triangulaire étendu sur la Nature et un dieu de l'orage, Pensée fécondante et Grand Architecte ; en second un doublet de bovins, des « grosses bêtes » localement équivalentes à des cachalots et aux parents de Gargantua, rois, prêtres et maîtres d'oeuvre ; en troisième rang un géant solitaire, factotum : Gargantua pour nous, cachalot divin autrefois. Un syncrétisme entre la culture à bovins du Villeneuve-Saint-Germain et une religion naturaliste reposant sur des grands cétacés s'est exporté dans presque toute la France. Conservée fidèlement dans le peuple pendant 7000 ans elle ressurgit presque intacte à la Renaissance. Les religions successives modifient la hiérarchie des rangs, perturbent les paires, changent les genres mais les entités : grande déesse, (avec ou sans représentation trinitaire), dieu de l'orage, prêtres, ouvriers survivent. Un des ultimes avatars, Saint-Michel, héritier de la Trinité géographique féminine défend l'unité de Dieu le Père (quis ut deus ?) au sein d'une Sainte-Trinité.

Kunst zwischen Körperlichkeit und Abstraktion zeigt sich exemplarisch in dieser Ausstellung zu Kunst in Zürich. In abstrakter Kunst nach Erotik und Poesie zu fragen und in körperlich obszöner Kunst nach kritischen Strategien, lässt auch ein Drittes Element zu, die Mystik, und damit ein Dreieck der Liebe. (Einführung)

This presentation applies the principles of triangular composition to create geometric art.

Instruction (April 15, 2022):
This presentation focuses on triangular composition, applying geometric shapes to 3 different images with mostly the use and creation of triangles. Summarized the circular flower petal image on a regular graphical coordinate plane from lesson 1 with a final image of what it is supposed to look like provided in slide 2, then we reviewed basic triangle properties including the Pythagorean Theorem, acute/obtuse angles, scalene, equilateral, isosceles, 45-45-90 triangles and 30-60-90 triangles. I then showed an image of a Canadian landscape where a Renaissance artist depicted the landscape of trees, an open plane and mountains in the background with geometric art. This is a more complex example of how to create geometric art with 2-D triangles. You could create a mountain range, with color or without. Explained how this type of art is considered "triangular composition". We then created our first drawing of different types of triangles on a white canvas plane. We wanted them overlapping, different sizes, diluted or translated, and then colored in if the students preferred. The purpose was to apply the concepts of different types of triangles and learn how to draw and overlap them with a straight-edge ruler and pencil. The next drawing was one where we created a fox or wolf-type face image with triangles and quadrilaterals. I walked the students through how to do this, with both the screenshare of the power point and my video, as well as instructional voice directions. The final image was the hardest and most complex, and applied the principles from the first two drawings. We created a compass circle, with an inscribed hexagon inside by creating arcs and chords along the outside of the circle, and working our way inward. A student asked about whether we could draw radii and/or diameters from the center to help them create equidistant angles connecting to the outside of the circle, and I said no, but for students who do not have the math compass, this could help create the equidistant angles of the hexagon inside the circle. I just recommend making the straight lines very light, so as not to distract away from the hexagons. Overall, the students learned to create anywhere between 10 - 15 hexagons inside the circle. This was an inscribed hexagon because it had 6 equal side lengths. If it was 5 sides, it would have been a pentagon. If it was 7 sides, it would have been a heptagon. Students also had the option to color in their image between the different layers, with color or black-and-white, if they chose to. A complimentary about 7 minute youtube video was provided to demonstrate the steps to making this circular image with link provided in the chat box. Some calming Jazz music was played in the background to create ambience. Students also had an opportunity to discuss how the pictures made them feel. Two students provided their thoughts here, by saying the final image made them feel like they were falling or going down a tunnel.

Link to instructional video on creating colorful inscribed hexagons inside a circle: https://youtu.be/32Lbvjiu7JQ

Reference Sources:

https://mymodernmet.com/polygon-art-landscape-paintings-elyse-dodge/
https://artradarjournal.com/2021/11/17/how-does-triangular-composition-create-balance-and-harmony-in-art/

Any cyclic quadrilateral whose sides are not parallel can define a triangle with one vertex at the point of intersection of the quadrilateral's diagonals and the other vertices at the points of intersection of the continuations of the quadrilateral's pairs of opposite sides. Using a cyclic quadrilateral and this triangle, the following four circles may be defined: the circumcircle of the quadrilateral, two circles with diameters that are the sides of the triangle that issue from the point of intersection of the quadrilaterals diagonals, and the Euler circle of the triangle. In the current paper, we shall prove four properties that relate the quadri-lateral, the triangle and these circles. We shall show that the two circles whose diameters are the sides of the triangle are perpendicular to the circumcircle of the quadrilateral. We shall prove equalities that relate the angle between the midlines in the quadrilateral with other angles. We shall show that the point of intersection of the midlines of the quadri-lateral belongs to the Euler's circle of the triangle defined using the quadri-lateral. In proving these properties we shall make use of the method of complex numbers in plane geometry, thereby illustrating different uses of this method of proof.

In this exploratory case study, we describe how teacher candidates with minimal mathematics knowledge of spherical geometry construct, discuss, manipulate and share the three-right angled triangle on a sphere using GeoGebra software. The study aims to demonstrate how mathematical understanding can begin with an object construction and manipulation in Dynamic Geometry Environment (DGE). Our ideas emerge from teacher candidates' use of the touchscreen-based computers mounted in a concept classroom following by embedding constructions in Canvas– Learning Management System (LMS), aiming to develop their understanding of spatial ability and spherical geometry. In this paper, we are not focusing on trigonometric laws, or other mathematical properties of the spherical triangle, instead we are introducing the construction of spherical triangle to not mathematics or science major teacher candidates. Some theory and techniques, including angel measurements on the sphere in a dynamic way are also learned along the way. Adapting Arzarello et al.'s (2014) theoretical framework of touch interactions and using Nemirovsky et al.'s (2013) perceptuomotor integration approach, augmented by a focus on the role of the bodily interactions in the mathematical learning with a touchscreen device, we assess and analyse how mathematical learning occurs when learners interact with touchscreen-based technologies and their peers. The result of the study showed that the mathematical construction and manipulation along with touchscreen-based interactions play a crucial role in developing spatial skills, which consequently triggers dynamic forms of mathematical communication. We also present evidence of collaborative engagement where teacher candidates engage in angle measurement of spherical triangles.

Considering the growing interest and approach in China to the courses of E/LE, the current study analyses the existing materials in market for teaching the History of Spanish Literature to foreign students and it also reects on Chinese culture of learning and on the knowledge that has Chinese students have about Western culture. From the data collected, the paper tries to provide some guidelines to develop a methodology structured by task-based communicative approach, respecting the instructions of the Plan Curricular del Instituto Cervantes and the Common European Framework of Reference for Languages for conducting a course on the History of Spanish Literature adressed to Chinese students. Finally, the study proposes a teaching unit that establishes synergies between the two methods of teaching, the Western communicative approach and Chinese culture of learning, with the aim of meeting the specic needs of students who are adressed to, and with the aspiration of oering a methodology...

The analysis of the correspondence between two sixteenth/seventeenth century female aristocrats —Luisa de Carvajal, Magdalena de San Jerónimo— enables us to reect on the value of relics and their relevance in the Spanish Baroque Counter-Reformation. We correlate Luisa's yearning for martyrdom as well as her mystical and religious exaltation with some of Magdalena's commitments at the Penitents House, at the Flemish Court, and as founder of the prison for prostitutes and female delinquents known as the Casa de la Galera, and, therefore, with dierent bodies and semiotics: the ill body of men, that polluting of 'working girls'; the bodies of virgins, of prostitutes, of actresses. These relate to dierent public spaces: the brothel, the theatre, the Magdalene asylum. And, in a very special manner, to the res publica and the Spanish Empire. All this based on the continuous background of Mary Magdalene's myth, a fundamental text that unites all the women considered here.

For the triangle ABC let (x,y)ABC denote the line which intersects the union of segments AB and BC in X and the segment AC in Y in a particular ratio (given in the file). Prove that the three lines (x,y)ABC, (x,y)BCA, (x,y)CAB intersect in a point dividing the segment NI in the ratio x:y, where N is the Nagel point and I is the incentre of ABC.

In this exploratory case study, we describe how teacher candidates with minimal mathematics knowledge of spherical geometry construct, discuss, manipulate and share the three-right angled triangle on a sphere using GeoGebra software. The study aims to demonstrate how mathematical understanding can begin with an object construction and manipulation in Dynamic Geometry Environment (DGE). Our ideas emerge from teacher candidates’ use of the touchscreen-based computers mounted in a concept classroom following by embedding constructions in Canvas– Learning Management System (LMS), aiming to develop their understanding of spatial ability and spherical geometry. In this paper, we are not focusing on trigonometric laws, or other mathematical properties of the spherical triangle, instead we are introducing construction of spherical triangle to not mathematics or science major teacher candidates. Some theory and techniques, including angel measurements on sphere in a dynamic way are also lear...