Terry's Maths Collections

Terry Tam, Mar 17, 2016

Table of Contents

  1. It's all about triangles...
    1. How to reconstruct a quadrilateral farm to a triangular one with the same area?
    2. Why same areas?
    3. Why ratio difference is the same?
    4. Why same areas? (2)
    5. Why same areas? (3)
    6. Shortest-perimeter Inscribed Triangle
    7. Right triangle with median as a geometric mean
    8. Shortest fence dividing a triangular field into halves
    9. 4 Centres in a Triangle
    10. Incircle and Excircles
    11. Inner/Outer Triangles
    12. Equilateral Triangle on 3 Concentric Circles
    13. Why 6 points are concyclic?
    14. Morley Equilateral Triangles
    15. Napoleon Equilateral Triangles
  2. It's all about cyclic quadrilaterals...
    1. Intersecting Chords Theorem and Secant-Tangent Theorem
    2. How to construct a bicentric quadrilateral with 3 points?
    3. Equilateral triangles on a triangle
    4. Simson's line
    5. Where am I?
    6. Why always a circle tangent?
    7. Ptolemy Theorem
  3. Famous Puzzle Gallery
    1. Eight Queens Puzzle
    2. right triangle paradox
  4. Origami
    1. Obtaining Fractions via Folding Paper
  5. Proofs without words
    1. triangular numbers-1
    2. The Sum of the Cubes of the First N Natural Numbers
    3. Centered Polygonal Number 中心多邊形數
    4. The Pythagorean Theorem.
    5. Proof without words : Area of the triangle
    6. Fibonacci Numbers and the Golden Spiral - demonstration
    7. Infinite Geometric Series
    8. Viviani's Theorem
    9. Fagnano's Problem
    10. QM-AM-GM-HM Inequality
  6. Elegant Solution
    1. All about Pythagoras theorem...
    2. What is the length of the side of equilateral triangle?
  7. Art Gallery
    1. Envelope and Complete Graph
    2. Modular Times Table
    3. Kaleidoscope
  8. Animation
    1. Petr-Douglas-Neumann Theorem

1.

It's all about triangles...

  • 1. How to reconstruct a quadrilateral farm to a triangular one with the same area?
  • 2. Why same areas?
  • 3. Why ratio difference is the same?
  • 4. Why same areas? (2)
  • 5. Why same areas? (3)
  • 6. Shortest-perimeter Inscribed Triangle
  • 7. Right triangle with median as a geometric mean
  • 8. Shortest fence dividing a triangular field into halves
  • 9. 4 Centres in a Triangle
  • 10. Incircle and Excircles
  • 11. Inner/Outer Triangles
  • 12. Equilateral Triangle on 3 Concentric Circles
  • 13. Why 6 points are concyclic?
  • 14. Morley Equilateral Triangles
  • 15. Napoleon Equilateral Triangles

1.1.

How to reconstruct a quadrilateral farm to a triangular one with the same area?

1. Given a quadrilateral farm ABCD, with its area unknown, how to reconstruct a new triangular one with a common edge which has the same area of ABCD?[br]2. How many such triangular farms can be reconstructed for a given quadrilateral farm?[br]3. How about the quadrilateral is concave?
– Terry Tam

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

1.7.

–

1.8.

–

1.9.

–

1.10.

–

1.11.

–

1.12.

–

1.13.

–

1.14.

–

1.15.

–

2.

It's all about cyclic quadrilaterals...

  • 1. Intersecting Chords Theorem and Secant-Tangent Theorem
  • 2. How to construct a bicentric quadrilateral with 3 points?
  • 3. Equilateral triangles on a triangle
  • 4. Simson's line
  • 5. Where am I?
  • 6. Why always a circle tangent?
  • 7. Ptolemy Theorem

2.1.

Intersecting Chords Theorem and Secant-Tangent Theorem

AB is a chord passing through P on a circle. It is trivial that when P is at the center of the circle, the product of lengths PA and PB (ie. the area of the rectangle) is the same for all possible diameters AB.[br]a) Prove that when P is not at the center of the circle, all possible chords AB form same-area rectangles. [Hint: Move point A to consider another chord passing through P][br]b) How about P is outside the circle?[br]
– Terry Tam

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

2.7.

–

3.

Famous Puzzle Gallery

  • 1. Eight Queens Puzzle
  • 2. right triangle paradox

3.1.

Eight Queens Puzzle

This is the famous eight queens puzzle. Try to place eight queens on the chessboard so that no two queens are on the same row, column, or diagonal. Just click on a square to place a queen or to remove it from the board. Good luck!
– David Radcliffe

3.2.

–

4.

Origami

  • 1. Obtaining Fractions via Folding Paper

4.1.

Obtaining Fractions via Folding Paper

Halves (1/2) can be simply found by folding a square paper. However, finding tri-sections (1/3 and its multiples) may not be trivial.[br]a) Using Haga's First Fold, drag point E (pretending corner B after folding) to point M (midpoint of the opposite side of AB) to see how tri-sections can be obtained. Could you explain why?[br]b) Using Haga's Third Fold, drag point E along the opposite side so that point J coincides with point N (midpoint of the opposite side of BC) to see another way to obtain tri-sections. Could you explain why?[br]c) Using Haga's First Fold, drag point E to quadrisection (1/4) of the opposite side of AB to see how to quintsect (5-sect) a line (and also its multiples). Do you find how to 7-sect? Could you generalize for other fractions as well?[br]d) As the result of part a), how 6-sections and 8-sections can be obtained?[br]e) As the result of part b), how 9-sections can be obtained? Do you find another way to get 6-sections as well?
– Terry Tam

5.

Proofs without words

  • 1. triangular numbers-1
  • 2. The Sum of the Cubes of the First N Natural Numbers
  • 3. Centered Polygonal Number 中心多邊形數
  • 4. The Pythagorean Theorem.
  • 5. Proof without words : Area of the triangle
  • 6. Fibonacci Numbers and the Golden Spiral - demonstration
  • 7. Infinite Geometric Series
  • 8. Viviani's Theorem
  • 9. Fagnano's Problem
  • 10. QM-AM-GM-HM Inequality

5.1.

triangular numbers-1

triangular numbers-1
– sonom

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

5.9.

–

5.10.

–

6.

Elegant Solution

  • 1. All about Pythagoras theorem...
  • 2. What is the length of the side of equilateral triangle?

6.1.

All about Pythagoras theorem...

ABCD is a square with length 13. ADE and BGC are triangles with length 5 and 12 respectively. What is the length of EG?
– Terry Tam

6.2.

–

7.

Art Gallery

  • 1. Envelope and Complete Graph
  • 2. Modular Times Table
  • 3. Kaleidoscope

7.1.

Envelope and Complete Graph

Under a necklace of n points...[br][math]K_n[/math]=complete graph[br][math]E_{n,r}[/math]=family of straight lines connecting points with interval of r points
– Terry Tam

7.2.

–

7.3.

–

8.

Animation

  • 1. Petr-Douglas-Neumann Theorem

8.1.

Petr-Douglas-Neumann Theorem

A hexagon (points A to F) has been prebuilt as example. Click "Generate" button to start the below animation:[br]1. For n-sided polygon, calculate n-2 consecutive angles with common difference = 360°/n.[br]2. For each side of polygon, create an isosceles triangle with apex angle = the angle used in this iteration.[br]3. If angle = 180°, the apex of the "hypothetical" isosceles triangle is the midpoint of the side of the polygon.[br]4. Connect the newly-created apexes as the new polygon and repeat step 2-4 until all angles are consumed.[br]5. Finally a regular n-sided polygon is created.[br][br]While animating...[br]1. try to move any points to transform the initial polygon. Can another regular n-sided polygon still be created?[br]a) initial polygon becoming a concave polygon[br]b) initial polygon becoming a disconnected polygon[br]2. try to toggle "outside" flag to "flip" the direction of the above isosceles triangle creation. Can another regular n-sided polygon still be created?[br]3. try to "shuffle" to consume the angles in different order. Can a regular n-sided polygon still be created? Any special of this polygon as compared with previous one?[br][br]Is such property just conserved for hexagons only? Click "Clear" button to clear the initial polygon and use the "Polygon" tool to construct your own.[br][br]P.S. Sorry about the "oscillating" animation behavior after the final regular n-sided polygon is created. Anyway, please stay tuned.
– Terry Tam

Information