GeoGebra in Visualizing some Chapters of Mathematics

Körtesi Péter, Oct 8, 2017

[b]Innovation in Engineering Education IEE 2025, [/b] - For tomorrow's engineers - Debrecen 7-8 February 2025. Faculty of Engineering at the University of Debrecen, the Faculty of Mechanical Engineering and Informatics at the University of Miskolc, and the Engineering Committee of the Regional Committee of the Hungarian Academy of Sciences

Table of Contents

  1. Elemi geometria -Elementary Geometry
    1. All triangles are equilateral. Is this true?
    2. Not all triangles are equilateral! See the right construction!
    3. Five circle problem
    4. Napolean triangle created with Lucie in Ostrava
    5. Isotomial transforms
    6. Isogononal transformations - image of a circle
    7. Isogonal transforms
  2. Függvények - Functions
    1. Definition of a function
    2. The inverse relation is NOT a function
    3. Two functions are "almost" equal, they differ only in x=1
    4. The exponential and logarithmic function base a and 1/a..
    5. Transformations power function, x^a
    6. The sinh(x) and its inverse.
    7. The symmetry of parabola order 3.
    8. Rac-31
    9. rac-2
    10. Inverse of the parabola
    11. The complete discussion of function x ln(x)
    12. A special function which has a limit in x=0
    13. Points of the Dirichlet function
    14. Definition of a real function - test
    15. Definition of a function
  3. Különleges görbék - Special curves
    1. Witch of Agnesi -Visualisation of Special curves in MacTutor History of Mathematics archive
    2. Pearls of Sluze- MacTutor visualisation
    3. Epitrochoid
    4. Visualising the Viviani curve
    5. Ellipszis és hiprbola - Ellipse and hyperbola
    6. Kardioidok- Cardioids
    7. Freshmen konzultáció: Cikloisok - Cycloids
    8. Fóliumok- Foliums
    9. Freshmen konzultáció: Simulókör -Osculating circle
    10. Conchoid - practical lesson in Rijeka Polytechnic
  4. Elemi matematika -Elementary Mathemtics
    1. Adding integers
    2. Multiplying fractions less than a unit
  5. Nem-Euklidészi geometriák modelljei Models of Non-Euclidean Geometries
    1. Poincaré model
    2. Half plane model, null triangle
    3. Half plane model
    4. Beltrami-Klein model
  6. Feladatlapok összeállítása -How to make up a simple exercice
    1. Test: Orthocentre of the triangle
    2. Central homothety
    3. Pyramid sections
    4. Constructions by Budai László
    5. Visualising simple solids
    6. Szabályos testek
    7. Visualising the Viviani curve
  7. Mérnöki matematika - Engineering Mathematics
    1. Mérési adatok közelítése polinommal- Freshmen konzultáció segédanyaga
    2. Mérési adatok közelítése a Freshmen konzultációk keretében -2
  8. Csapatverseny - GeoGebra Team Competition
    1. GeoGebra team competition 2. problem
    2. GeoGebra Team Competition 1 problem
    3. GeoGebra Team Competitions - First round, Problem 3.

1.

Elemi geometria -Elementary Geometry

  • 1. All triangles are equilateral. Is this true?
  • 2. Not all triangles are equilateral! See the right construction!
  • 3. Five circle problem
  • 4. Napolean triangle created with Lucie in Ostrava
  • 5. Isotomial transforms
  • 6. Isogononal transformations - image of a circle
  • 7. Isogonal transforms

1.1.

All triangles are equilateral. Is this true?

In triangle ABC the bisector AF and mediator DF of side BC meet in F. The triangles T_1 and T_2, moreover triangles H_1 and H_2 are equal, thus the sides AB and AC of the triangle ABC are of equal lengths. Repeating for the other sides, e.g AB and BC, w
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1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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2.

Függvények - Functions

  • 1. Definition of a function
  • 2. The inverse relation is NOT a function
  • 3. Two functions are "almost" equal, they differ only in x=1
  • 4. The exponential and logarithmic function base a and 1/a..
  • 5. Transformations power function, x^a
  • 6. The sinh(x) and its inverse.
  • 7. The symmetry of parabola order 3.
  • 8. Rac-31
  • 9. rac-2
  • 10. Inverse of the parabola
  • 11. The complete discussion of function x ln(x)
  • 12. A special function which has a limit in x=0
  • 13. Points of the Dirichlet function
  • 14. Definition of a real function - test
  • 15. Definition of a function

2.1.

Definition of a function

Visualizing the way of the inverse mapping of the real function f(x)=x^2 for the interval [1,6].
It is easy to see that the restriction of the function on the interval [1,6] and its codomain on the interval [1,10] it is not an injection, but a surjection.
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2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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2.11.

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2.12.

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2.13.

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2.14.

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2.15.

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3.

Különleges görbék - Special curves

  • 1. Witch of Agnesi -Visualisation of Special curves in MacTutor History of Mathematics archive
  • 2. Pearls of Sluze- MacTutor visualisation
  • 3. Epitrochoid
  • 4. Visualising the Viviani curve
  • 5. Ellipszis és hiprbola - Ellipse and hyperbola
  • 6. Kardioidok- Cardioids
  • 7. Freshmen konzultáció: Cikloisok - Cycloids
  • 8. Fóliumok- Foliums
  • 9. Freshmen konzultáció: Simulókör -Osculating circle
  • 10. Conchoid - practical lesson in Rijeka Polytechnic

3.1.

Witch of Agnesi -Visualisation of Special curves in MacTutor History of Mathematics archive

The file is illustrating the evolute of the curve, following two ways, the centre of the osculating circle, and the convex boundary of the normals. In order to see the visualisation animate the parameter t. To change the shape of the initial curve, change a.
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3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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3.10.

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4.

Elemi matematika -Elementary Mathemtics

  • 1. Adding integers
  • 2. Multiplying fractions less than a unit

4.1.

Adding integers

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4.2.

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5.

Nem-Euklidészi geometriák modelljei Models of Non-Euclidean Geometries

  • 1. Poincaré model
  • 2. Half plane model, null triangle
  • 3. Half plane model
  • 4. Beltrami-Klein model

5.1.

Poincaré model

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5.2.

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5.3.

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5.4.

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6.

Feladatlapok összeállítása -How to make up a simple exercice

A GeoGebra alkalmas egyszerű vagy összetett dinamikus feladatlapok összeállításra is.

  • 1. Test: Orthocentre of the triangle
  • 2. Central homothety
  • 3. Pyramid sections
  • 4. Constructions by Budai László
  • 5. Visualising simple solids
  • 6. Szabályos testek
  • 7. Visualising the Viviani curve

6.1.

Test: Orthocentre of the triangle

[b]Magasságpont - Ortocentre[/b][b][br]Definíció:[/b] Az ABC háromszög magasságvonalainak H metszéspontját a háromszög magasságpontjának,[br][b]ortocentrum[/b]ának nevezzük.[br][b]Definition:[/b] The meet point of the three altitude line is called the ORTOCENTRE of the triangle[br][br][b]Feladat:[/b] Tanulmányozza a különböző[br]eseteket, és döntse el, mikor lehet a háromszög magasságpontja az adott[br]háromszög belső, vagy külső pontja, esetleg a háromszög kerületi pontja?[br][b]Problem[/b][br]using the sketch try to find if the ortocentre can be external point of the triangle- belonging to the outer part of the triange?
Can you find the condition for the ortocentre to coincide with one of the vertices?[br]Mikor esik egybe az ortocentrum és egy csúcsa a háromszögnek?
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6.2.

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6.3.

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6.4.

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6.5.

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6.6.

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6.7.

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7.

Mérnöki matematika - Engineering Mathematics

  • 1. Mérési adatok közelítése polinommal- Freshmen konzultáció segédanyaga
  • 2. Mérési adatok közelítése a Freshmen konzultációk keretében -2

7.1.

Mérési adatok közelítése polinommal- Freshmen konzultáció segédanyaga

Az fN függvény az N-ed fokú polinom közelítést jelenti (N=2,5,9,15,23,24).
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7.2.

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8.

Csapatverseny - GeoGebra Team Competition

  • 1. GeoGebra team competition 2. problem
  • 2. GeoGebra Team Competition 1 problem
  • 3. GeoGebra Team Competitions - First round, Problem 3.

8.1.

GeoGebra team competition 2. problem

Different solutions - Különböző megoldások
1. Official proposal: Using Ptyhagoras theorem[br]2. Vectors: Using the condition for perpendicularity of two vectors[br]3. Coordinate Geometry: writing the slopes of the straight lines[br]4. Using construction of circles [br]
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8.3.

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