از انیمیشن کمک بگیریم

 

قضيه سوا Ceva's Theorem

Ceva's Theorem. Concurrency. Interactive proof with animation. Key concept: Menelaus Theorem. Puzzle of the Ceva's Theorem: 48 classic piece اثبات قضيه مرحله به مرحله با انيميشن.

قضيه پروانه Butterfly Theorem

Butterfly Theorem Proof with animation اثبات قضيه مرحله به مرحله با انيميشن.

قضيه مورليMorley's Theorem

Morley's Theorem. Introduction with animation. Triangle + Trisectors = Equilateral triangle. Morley's Theorem Puzzle: 22 pieces of polygons اثبات وشرح قضيه مرحله به مرحله با انيميشن.

قضيه منلائوسMenelaus' Theorem

Menelaus' Theorem. Interactive proof with animation and key concepts اثبات قضيه مرحله به مرحله با انيميشن.

اجسام افلاطونیPlatonic Solids

Platonic Solids. Interactive animation

Eyeball Theorem

Eyeball Theorem: Animated Angle to Geometry Study.

Hexagon and Lissajous Art

Hexagon and Lissajous Art

 

اثبات مرحله به مرحله

Nagel Point Theorem Proof

Butterfly Theorem

Ceva's Theorem

Clifford's Theorems

Bevan Point

بعضی از نقاط خاص درمثلث

 1. Centroid

 2.  Orthocenter

 3.  Circumcenter

 4.  Incenter

 5.  Excenter

 6. Apollonius Point

7. Nine-Point Center

 8. Gergonne Point

 9. Nagel Point.

 10. Fermat Point

 11. Napoleon Point

 12. Bevan Point V. See: Illustration with animation

 13. Vecten Point V

 14. Symmedian (Lemoine) Point 

Symmedian (Lemoine) Point

 15. Mittenpunkt or Middlespoint 

Mittenpunkt Point

 16. Steiner Point

Steiner Point

مساله3

In triangle ABC, angle B is a right angle. BD is the altitude. r1, and r2, are the inradii of the triangles ADB, and BDC respectively, the points E, F, and G are points of tangency. Prove that .

مساله2

Prove the following: if, in a triangle ABC,  mÐ BAO = mÐ OAC = 20°, mÐ ACO = 10°,and mÐ OCB = 30°, then mÐ OBC = 80°.

مساله

Prove the following: if, in a triangle ABC, median BD is such that mÐ A = mÐ DBC, and mÐ ADB = 45°, then mÐ A = 30°.

مسائل وقضایای هندسه

Table of Content

1. Triangles. Theorems and problems about triangles.

2. Circles. Theorems and problems about circles.

3. Triangle Centers

4. Archimedes' Book of Lemmas

5. Quadrilaterals

6. Butterfly Theorem. See also: Butterfly Theorem Puzzle

7. Carnot's Theorem

8. Ceva's Theorem

9. Clifford's Circles Chain Theorems

10. Equal Incircles Theorem

11. Equilic Quadrilateral.

12. Eyeball Theorem. Illustration

13. Gergonne Point Theorem

14. Heron's Formula. Key facts and a purely geometric step-by-step proof.

15. Johnson's Theorem

16. Kurschak's Tile & Theorem

17. Langley Problem: 20° Isosceles Triangle Adventitious angles.

18. Menelaus' Theorem

19. Miquel's Pentagram. Proof

20. Miquel's Pentagram. Dynamic Geometry Cabri, GSP, Cinderella, C.a.R.

21. Monge & d'Alembert Three Circles Theorem I - Dynamic Geometry Requires Java applet 1.3 or higher.

22. Monge & d'Alembert Three Circles Theorem II - Dynamic Geometry Requires Java applet 1.3 or higher.

23. Morley's Theorem

24. Napoleon's Theorem

25. Nagel Point Theorem. See also: Nagel Point Flowchart Proof, Nagel Point Puzzle.

26. Newton's Theorem: Newton's Line

27. Parallelogram with Squares theorem.

28. Pentagons and Pentagrams. Menelaus and Collinearity

29. Platonic Solids. Interactive animation.

30. Poncelet's Theorem

31. Proposed Problems about congruence of line segments, angles, and triangles. Level: High School, SAT Prep, College geometry.

32. Sangaku Problem

33. Sangaku Problem 2

34. Sangaku Problem 3

35. Semiperimeter and excircles of a triangle

36. Semiperimeter and incircle of a triangle

37. Semiperimeter, incircle and excircles of a triangle

38. Seven Circles Theorem

39. Simson Line

40. Simson Line: Angle between two Simson lines

41. Steiner's Theorem

42. 100° Isosceles Triangle Problem

43. Triangles

44. Triangle with the bisectors of the exterior angles. Collinearity

45. Triangle & squares. Fifteen theorems, visual illustrations.

46. Triangle with Squares 0 Two squares.

47. Triangle with Squares 1 Two squares.

48. Triangle with Squares 2 Two squares.

49. Triangle with Squares 3 Three squares.

50. Triangle with Squares 4 Finsler-Hadwiger Theorem.

51. Triangle with Squares 5 Two squares, median and altitude.

52. Triangle with Squares 6 Four squares.

53. Triangle Centers

54. Van Aubel's Theorem Quadrilateral with Squares

55. Varignon and Wittenbauer parallelograms

56. Adjacent Angles and Five Bisectors

Naming and Classification of Curves

Classification of Curves

How Curves are Named

Other Constructions and Definitions

Curve Family Tree

Related Web Sites

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Classification of Curves

There are many ways to classify curves. One way is to determine whether a curve is the graph of some polynomial equation p[x,y]==0. The graph of a polynomial equation are called algebraic curves. Algebraic curve are assigned an order. The order of an algebraic curve is the degree of the polynomial. For example, line (a x + b y + c == 0); circle ((x+h)^2 + (y+k)^2 -r^2 == 0), or the deltoid ((x^2+y^2)^2 - 8 a x (x^2 - 3 y^2) + 18 a^2 (x^2 + y^2) - 27 a^4 == 0), are algebraic curves. Curves may be easy to trace but are not algebraic. For example, no polynomial's graph can be any of cycloid, equiangular spiral, or quadratrix of Hippias. Algebraic curves with degree greater than 2 are called higher plane curves. Non-algebraic curves are called transcendental curves. To determine whether a curve is algebraic requires graduate level math knowledge, and is beyond this project's scope. Here we will group curves by tracing methods.

There is a special class of curves known as fractal and space-filling curves. Simply, fractal curves are curves that are not smooth. All the curves covered here are such that when you keep magnifying parts of the curve, it'll eventually looks like a line, unless you are magnifying a cusp point. Fractal curves are such that no matter how large you magnify, the curve is still corrugated. In a sense, it's all cusps! Space filling curves are a special class of fractal curves, so named because they completely fill an area. Fractal and space-filling curves are discovered in late 17th century and their existence shocked mathematicians profoundly. They are interesting as a topic but this project will not deal with fractal or space-filling curves. Space-filling curves are also known as Peano curve, after logician Giuseppe Peano. Famous fractals includes the Helge von Koch's "Koch snowflake", David Hilbert's spacing-filling curve, William Gosper's "flowsnake", Benoit Mandelbrot's Mandelbrot set, George Canter's "Canter Dust", and others. (see Related Web Sites)

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How Curves are Named

Here are some examples of how curves got named. The list is not exhaustive.

By Person's names: quadratrix of Hippias, conchoid of Nicomedes, Kappa curve, limacon of Pascal, lemniscate of Bernoulli, witch of Agnesi, folium of Descarte, trisectrix of Maclaurin, trident of Newton (aka parabola of Descartes), Bowditch curve, Tschirnhausen's cubic (aka trisectrix of Catalan, L'Hospital's cubic), Cayley's sextic, nephroid of Freeth. Euler's spiral (aka Clothoid ), Euler's curve (x^y==y^x), spiral of Cornu, Durer's conchoid, lissajous (aka Bowditch curve).

Note that in general, the name attached to a curve or a math theorem is not necessary the person who invented or worked on it.

By method of construction: caustics, derivative, integral, isoptic, orthoptic, parallel, pedal, radial, envelope, evolute, involute, cissoid, conchoid, roulette, glissette, strophoid, pursuit curve.

By property: trisectrix, quadratrix, brachistochrone (aka cycloid), tautochrone (aka cycloid), isochrone (aka semi-cubic parabola, semi-cubical parabola), anallagmatic curve.

By shape: astroid (star), deltoid (greek letter Delta) (aka tricuspid, Steiner's hypocycloid), cardioid (hear-shaped), conchoid of Nicomedes (mussel-shaped) (aka cochloid), nephroid (kidney-shaped), cycloid (circle, wheel), folia (leaf), trident of Newton (aka parabola of Descartes), serpentine (snake) (named by Newton), cissoid of Diocles (Ivy-shaped), rose (aka rhodonea), catenary (chain-shaped) (aka chainette, alysoid), tractrix (aka equitangential curve), spiral, lemniscate of Bernoulli (ribbon-shaped), lituus (crook), figure eight curve (aka lemniscate of Gerono), bullet nose, cross curve.

Historical reasons: ellipse, parabola, hyperbola, witch of Agnesi, limacon of Pascal (snail of Pascal, originated from Roberval).

By the form of the formula: semi-cubic parabola (aka isochrone, semi-cubical parabola), Parabola of Descartes (?) (aka trident of Newton).

Not exactly sure (research to do): circle, line, right strophoid (Barrow, 1670), Kampyle of Eudoxus, Kappa curve (Gutschoven's curve), Hippopede (horse fetter) (Proclus, ca.75 BC), bicorn (Sylvester, 1864), piriform (pear-shaped quartic) (De Longchamps, 1886), Clothoid, cochleoid (bernoulli, 1726), Cayley's sextic, Devil's curve (Cramer, 1750).

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Other Constructions and Definitions

Brachistochrone (from Greek, brakhus:short, chrone:time) means a curve that connects to two given points such that a particle sliding from the higher point to the lower point under ideal physical law (ideal gravitational force, no friction, no air-resistance, particle has no volume ...etc.) will descent with the fastest time, among all possible curves. The unique solution is the cycloid, not straight line.

Tautochrone (Greek, tauto:same, chrone:time) is similar to brachistochrone. It's a curve that, connects two given points such that it takes the same amount of time for a particle to slide from any point on the curve to the lower point, under ideal physical law. This may sound paradoxical. Intuitively, the longer path the particle needs to glide, the longer time it will have to take. However, consider that acceleration rate is high when the slope is sharp, therefore it is possible for a curve whose curvature balances out so that no matter where the particle starts, it always takes the same time to reach the end. The unique solution is the cycloid.

The word Isochrone (Greek, iso:equal, chrone:time) is sometimes used to mean tautochrone, but other authors use it to mean the semi-cubic parabola -- a curve which a particle will descend equal vertical distances in equal time intervals, under ideal physical law. The latter meaning will be used here.

Anallagmatic curve is any curve that has inversion that's itself. Circle and Cassinian oval are examples.

Isoptic of a given curve C and a given angle α is the locus of a point P such that P is the intersection of tangents of C that meets in angle α. If α:=π/2, then the derived curve is called an orthoptic. For example, the orthoptic of a parabola is its directrix, and the orthoptic of an ellipse, hyperbola, or deltoid is a circle.

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Curve Family Tree

The phrases in parenthesis are reminders. Not all curves in this project appears below.

• roulettes (curve rolling on curve)
  • cycloidal curves (circle rolling on circle/line)
    • epitrochoids (circle rolling outside another circle)
      • circle
      • epicycloids (tracing point on circle)
        • cardioid (1 cusp)
        • nephroid (2 cusps)
      • roses (r==Cos[p/q*θ])
        • circle
        • trifolium (3 petals)
        • quadrifolium (4 petals)
    • hypotrochoids (circle inside a circle. Either circle may be the rolling circle.)
      • hypocycloids (tracing point on circle)
        • ellipses, line segment
        • deltoid (3 cusps)
        • astroid (4 cusps)
        • cardioid (by double generation)
        • nephroid (by double generation)
    • trochoids (circle rolling on a line)
      • line (tracing point on center of circle)
      • prolate (extended) cycloids
      • curtate (contracted) cycloids
      • cycloid (tracing point on circle)
• conic sections (r==e/(1+e*Cos[θ]))
  • e>1, hyperbolas
  • e==1, parabola
  • e<1, ellipses
• Cassinian ovals (locus of points whoes product of distances to two fixed points is a constant)
  • lemniscate of Bernoulli (two loops with a double point in center)
• conchoids (shifting the curve by a constant to a point on radial line from a pole.)
  • limacon of Pascal (conchoids of a circle)
    • cardioid
    • trisectrix
  • Conchoids of Nicomedes (conchoids of a line)
• cissoids (distance of two curves by a radial line from a pole)
  • cissoid of a Line and a Circle
    • oblique cissoid
      • cissoid of Diocles
    • strophoid
  • circle
  • line
  • conchoid of Nicomedes
  • lemniscate of Bernoulli
• spirals
  • Archimedean spiral (r == θ^n)
    • (n:=1), Archimedes' spiral
    • (n:=1/2), parabolic spiral (aka Fermat's spiral)
    • (n:=-1), hyperbolic spiral (aka reciprocal spiral)
    • (n:=-1/2), lituus
    • (n:=-0), circle
  • equiangular spiral (r == E^(θ*Cot[α]))
    • (α:=π/2), circle
  • Euler's Spiral (aka Clothoid, Cornu's spiral)
• sinusoidal “spirals” (r^n==a^n*Cos[n*θ], n rational)
  • (n:=-2), rectangular hyperbola
  • (n:=-1), line
  • (n:=-1/2), parabola
  • (n:=-1/3), Tschirnhausen's cubic
  • (n:=1/3), Cayley's Sextic
  • (n:=1/2), cardioid
  • (n:=1), circle
  • (n:=2), lemniscate of Bernoulli
• lissajous
  • ellipse

Sinusoidal “spiral” is defined to be r^n==a^n*Cos[n*θ], n rational. It is so-called because it's a “spira” but the radius vector increases and decreases “sinuously”.

Strophoid

Base Curve	Pole	Fixed Point	Strophoid
line	not on line	on line	oblique strophoid
circle	center	on curve	nephroid of Freeth

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Related Web Sites

see Generic Reference Page.

Fractals

Many visual representations of iterative process are called fractals, not all of which are remotely considered curves. There are hundreds fractals resources on the web. Here are few selected ones.

William McWorter. A tutorial on L-systems -- an iterative process that can generate space-filling curves and other fractals. (7 long pages. excellent) http://spanky.triumf.ca/www/fractint/lsys/tutor.html

FractInt has a page that explains various types of fractals. http://spanky.triumf.ca/www/fractint/fractal_types.html. FractInt is an excellent (free) fractal program by coorperative efforts of programers.

Non-linear science FAQ, explains fractals, chaos and others. (excellent) http://www.faqs.org/faqs/sci/nonlinear-faq/

Robert M. Dickau has graphics on fractals and chaos. http://mathforum.org/advanced/robertd/index.html#frac Especially the page on space-filling curves. http://mathforum.org/advanced/robertd/lsys2d.html.

MacTutor short biographies on Giuseppe Peano, Helge von Koch, Benoit Mandelbrot, Gaston Maurice Julia.

Curves

MacTutor Famous Curve Index on Plane Curve terminology.

Silvio Levy/CRC Press. Geometry Formulas and Facts. http://www.geom.umn.edu/docs/reference/CRC-formulas/ In particular the page on Special Plane Curves. http://www.geom.umn.edu/docs/reference/CRC-formulas/node32.html

★ Back to Table of Contents

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Content updated: 1999.
Last modified: 2002-03-28.
© 1995-97 by Xah Lee. ([e-mail])
http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html