Encyclopedia of Geometry TABLE OF CONTENTS

1.    Section1. Arising geometry , The role of geometry in science, industry, and . . .

1.1.   Arising geometry  and its evolution

1.2.   The role of geometry in mathematics

1.3.   Applying geometry in science, industry and life

1.4.   The role of geometry in reinforcing power of thought

1.5.   How to solve a geometry problem?

2.    Section 2. Point, Line, Angle

2.1.   Definition and Theorem

2.2.   Point

2.3.   Line segment, Line

2.3.1.    Number of segments, Number of lines

2.3.2.    Measure of a segment

2.3.3.    inequality of segments

2.3.4.    Other problems relating to this part

2.4.   Angle

2.4.1.    Measure of an angle

2.4.2.    Relations among angles

2.4.3.    Other problems relating to this part

2.4.4.    Combined Problems

3.    Section 3. Lines which are:  parallel, Perpendicular, bisector, . . .

3.1.   Definition and Theorem

3.2.   Angle

3.2.1.    Measure of an Angle

3.2.2.    Relations among angles

3.3.   (line) Segment

3.3.1.    Relations among segments

3.4.    Lines which are: Parallel, Perpendicular, . . .

3.4.1.    The lines are parallel

3.4.2.    The lines are perpendicular

3.4.3.    The line is a bisector

3.5.   Other problems relating to this part

4.    Section 4. Triangle

4.1.    Triangle of any type

4.1.1.    Definition and Theorem

4.1.2.    Angle

4.1.2.1.     Measure of an angle

4.1.2.1.1.    Measure of the angles of a given triangle

4.1.2.1.2.    Measure of the angles of triangles and other shapes

4.1.2.2.     Relations among angles (equalities)

4.1.2.2.1.      Relations among Angles (equalities)

4.1.2.2.2.      Relations among Angles (in equalities)

4.1.3.    side

4.1.3.1.     Measure of a side

4.1.3.2.     The interval of a side

4.1.3.3.     Other problems relating to this part

4.1.4.    Altitude, median, bisector

4.1.4.1.     Altitude

4.1.4.2.     median

4.1.4.3.     bisector

4.1.5.    (line) segment

4.1.5.1.     length of a segment

4.1.5.2.     relations  among segments

4.1.5.2.1.      relations  among segments (equalities)

4.1.5.2.2.      relations  among segments (inequalities)

4.1.5.3.     other problems relating this section

4.1.6.    perimeter

4.1.6.1.     length of perimeter

4.1.6.2.     relations among perimeters

4.1.6.2.1.      relations among perimeters (equalities)

4.1.6.2.2.      relations among perimeters (inequalities)

4.1.6.3.     other problems relating to this part

4.1.7.    Area

4.1.7.1.     measure of area

4.1.7.1.1.      measure of area of a given triangle

4.1.7.1.2.      measure of area of triangles and other shapes

4.1.7.2.     ratio of areas

4.1.7.3.     a relation about areas

4.1.7.4.     equivalent triangles

4.1.7.5.     other problems relating to this part

4.1.8.      congruent triangles

4.1.9.      special points

4.1.10.   collinear points

4.1.11.   concurrent lines

4.1.12.   lines which are: parallel, perpendicular, . . .

4.1.12.1.   the lines are parallel

4.1.12.2.   the lines are perpendicular

4.1.12.3.   the line is bisector

4.1.12.4.   the lines passes through a fixed point

4.1.12.5.   other problems relating to this part

4.1.13.   created figures

4.1.13.1.   created figures (triangles)

4.1.13.2.   created figures (polygons)

4.1.14.   other problems relating to triangles of any type

4.1.15.   mixed problems

4.2.   equilateral triangle

4.2.1.    definition and theorem

4.2.2.    angle

4.2.2.1.     measure of an angle

4.2.2.2.     the relation among angles

4.2.3.    side

4.2.3.1.     length of a side

4.2.4.    altitude, median, bisector

4.2.4.1.     length of altitude

4.2.5.    (line) segment

4.2.5.1.     length of a segment

4.2.5.2.     relation among segments

4.2.6.    perimeter

4.2.6.1.     length of perimeter

4.2.6.2.     relations among perimeters

4.2.7.    area

4.2.7.1.     measure of area

4.2.8.      congruency of triangles

4.2.9.      prove that a triangle is equilateral

4.2.10.   other problems relating to this part

4.3.   Isosceles triangle

4.3.1.    Definition and theorem

4.3.2.    angle

4.3.2.1.     measure of angle

4.3.2.2.     relation between angles

4.3.2.2.1.      relations among angles (equalities)

4.3.2.2.2.      relations among angles (inequalities)

4.3.3.    side

4.3.3.1.     length of side

4.3.4.    altitude, median, bisector

4.3.4.1.     altitude

4.3.4.2.     median

4.3.4.3.     bisector

4.3.4.4.     other problems relating to this part

4.3.5.    (line)segment

4.3.5.1.     length of a segment

4.3.5.2.     relations among segments

4.3.5.2.1.      relations among segments (equlities)

4.3.5.2.2.      relations among segments (inequlities)

4.3.6.    perimeter

4.3.6.1.     length of perimeter

4.3.6.2.     area

4.3.6.3.     measure of area

4.3.7.      triangles congruency

4.3.8.      collinear points

4.3.9.      lines which are: parallel, perpendicular, . . .

4.3.9.1.      the lines are parallel

4.3.9.2.      lines are perpendicular

4.3.9.3.      Prove: the triangle is isosceles

4.3.9.4.      the given triangle is isosceles

4.3.9.5.      the other generated triangles are isosceles

4.3.10.   other problems relating to this part

4.3.11.   mixed problems

4.4.   right triangle

4.4.1.    definition and theorem

4.4.2.    angle

4.4.2.1.     measure of angle

4.4.2.2.     relations among angles

4.4.3.    side

4.4.3.1.     length of side

4.4.3.2.     relations among sides

4.4.3.2.1.      relations among sides (equalities)

4.4.3.2.2.      relations among sides (inequalities)

4.4.4.    altitude, median, bisector

4.4.4.1.     altitude

4.4.4.2.     median

4.4.5.    (line) segment

4.4.5.1.     length of segment

4.4.5.2.     relations among segments

4.4.5.2.1.      relations among segments ((equalities)

4.4.5.2.2.      relations among segments ((equalities)

4.4.6.    perimeter

4.4.6.1.     length of perimeter

4.4.7.    area

4.4.7.1.     measure of area

4.4.7.2.     relations among areas

4.4.8.      congruency of triangles

4.4.9.      collinear points

4.4.10.   lines which are: parallel, perpendicular, . . .

4.4.10.1.   the line is a bisector

4.4.10.2.  the line passes through a fixed point

4.4.11.   generated figures

4.4.12.   Prove: the triangle is right

4.4.13.   other problems relating to this part

4.4.14.   mixed problems

4.5.   acute (scalene) triangle

4.5.1.    definition and theorem

4.5.2.    angle

4.5.2.1.     measure of angle

4.5.3.    side

4.5.3.1.     length of side

4.5.4.    altitude, median, bisector

4.5.4.1.     length of altitude

4.5.5.    (line) segment

4.5.5.1.     relations among segments (inequalities)

4.5.6.    perimeter

4.5.6.1.     length of perimeter

4.5.6.2.     a relation between perimeters

4.5.7.    area

4.5.7.1.     measure of area

4.5.8.    generated figures

4.6.   obtuse (scalene) triangle

4.6.1.    definition and theorem

4.6.2.    angle

4.6.2.1.     measure of angle

4.6.3.    side

4.6.3.1.     length of side

4.6.4.    altitude, median, bisector

4.6.4.1.     bisector

4.6.5.    (line) segment

4.6.5.1.     relations among segments

4.6.5.1.1.      relations among segments (equalities)

4.6.5.1.2.      relations among segments (inequalities)

4.6.6.    perimeter

4.6.6.1.     length of perimeter

4.6.7.    area

4.6.7.1.     measure of area

4.6.8.      congruency of triangles

4.6.9.      collinear points

4.6.10.   generated figures

5.    Section 5.  Special quadrilaterals

5.1.   parallelogram

5.1.1.    definition and theorem

5.1.2.    angle

5.1.2.1.     measure of angle

5.1.2.2.     relations among angles

5.1.3.    side

5.1.3.1.     length of side

5.1.4.    diagonal

5.1.5.    (line) segment

5.1.5.1.     relations among segments

5.1.6.    perimeter

5.1.6.1.     length of perimeter

5.1.7.    area

5.1.7.1.     measure of area

5.1.7.2.     measure of areas of generated figures

5.1.7.3.     ratio of areas

5.1.7.4.     relations among areas

5.1.8.    congruency of parallelograms

5.1.9.    special points

5.1.10.   collinear points

5.1.11.   concurrent lines

5.1.12.   lines which are: parallel, perpendicular, . . .

5.1.12.1. the lines are parallel

5.1.12.2. the line is bisector

5.1.13.   generated figures

5.1.14.   Prove: the quadrilateral is a parallelogram. 

5.1.15.   other problems related to this part

5.1.16.   mixed problems

5.2.   rectangle

5.2.1.    definition and theorem

5.2.2.    angle

5.2.2.1.     measure of angle

5.2.2.2.     the relations among angles

5.2.3.    side

5.2.3.1.     length of side

5.2.3.2.     increase or decrease of side

5.2.4.    diagonal

5.2.5.    (line) segment

5.2.6.    perimeter

5.2.7.    area

5.2.7.1.     measure of area

5.2.7.2.     area of generated figures

5.2.7.3.     a relation among areas

5.2.8.    congruency of rectangles

5.2.9.    special points (center of symmetry of a rectangle)

5.2.10.   collinear points

5.2.11.   generated figures

5.2.12.   other problems related to this part

5.3.   square

5.3.1.    definition and theorem

5.3.2.    angle

5.3.2.1.     measure of angle

5.3.3.    side

5.3.3.1.     length of side

5.3.4.    diagonal

5.3.4.1.     length of diagonal

5.3.5.    (line) segment

5.3.5.1.     length of segment

5.3.5.2.     relations among segments

5.3.5.3.     relations among segments (equalities)

5.3.5.4.     relations among segments (inequalities)

5.3.6.    perimeter

5.3.6.1.     length of perimeter

5.3.7.    area

5.3.7.1.     measure of area

5.3.7.2.     area of generated figures

5.3.8.    congruency of squares

5.3.9.    special points

5.3.10.   collinear points

5.3.11.   concurrent lines

5.3.12.   lines which are: parallel T perpendicular, . . .

5.3.13.   generated figures

5.3.14.   Prove: the quadrilateral is a square

5.3.15.   other problems relating to this part

5.4.   rhombus

5.4.1.    definition and theorem

5.4.2.    angle

5.4.2.1.     measure of angle

5.4.3.    side

5.4.3.1.     length of side

5.4.4.    diagonal

5.4.4.1.     length of diagonal

5.4.5.    (line) segment

5.4.5.1.     length of segment

5.4.5.2.     relations among segments

5.4.6.    perimeter

5.4.6.1.     length of perimeter

5.4.7.    area

5.4.7.1.     measure of area

5.4.8.      congruency of rhombuses

5.4.9.      special points

5.4.10.   lines which are: parallel, perpendicular, . . .

5.4.10.1. the line is bisector

5.4.11.   generated figures

5.4.12.   Prove: the quadrilateral is a rhombus

5.4.13.   mixed problems

5.5.   trapezoid

5.5.1.    trapezoid of any type

5.5.1.1.     definition and theorem

5.5.1.2.     angle

5.5.1.2.1.      measure of angle

5.5.1.2.2.      relations among angles

5.5.1.3.     side

5.5.1.3.1.      length of side

5.5.1.3.2.      relations among sides

5.5.1.4.     diagonal

5.5.1.4.1.      length of diagonal

5.5.1.5.     (line) segment

5.5.1.5.1.      length of segment

5.5.1.6.     perimeter

5.5.1.6.1.      length of perimeter

5.5.1.7.     area

5.5.1.7.1.      measure of the area of trapezoid

5.5.1.7.2.      measure of the areas of the generated figures

5.5.1.7.3.      a relation relating to areas

5.5.1.8.       congruency of trapezoids

5.5.1.9.       collinear points

5.5.1.10.   concurrent lines

5.5.1.11.   lines which are: parallel, perpendicular,
  bisector, . . .

5.5.1.12.   Prove: the quadrilateral is a trapezoid

5.5.1.13.   other problems relating to this part

5.5.1.14.   mixed problems

5.5.2.    isosceles trapezoids

5.5.2.1.     definition and theorem

5.5.2.2.     angle

5.5.2.2.1.      measure of angle

5.5.2.3.     side

5.5.2.4.     diagonal

5.5.2.5.     (line) segment

5.5.2.6.     length of segment

5.5.2.7.     perimeter

5.5.2.7.1.      length of perimeter

5.5.2.8.     area

5.5.2.8.1.      measure of area

5.5.2.9.       congruency of isosceles triangles

5.5.2.10.   lines which are: parallel, perpendicular, bisector, . . .

5.5.2.10.1.  the lines are perpendicular

5.5.2.11.   Prove: the quadrilateral is a trapezoid

5.5.2.12.   other problems relating to this part

5.5.2.13.   mixed problems

5.5.3.    right (angle) trapezoid

5.5.3.1.     definition and theorem

5.5.3.2.     angle

5.5.3.2.1.      measure of angle

5.5.3.3.     (line) segment

6.    Section 6.  other quadrilaterals

6.1.   definition and theorem

6.2.   angle

6.2.1.    measure of angle

6.2.2.    relations among angles

6.2.2.1.     relations among angles (equalities)

6.2.2.2.     relations among angles (inequalities)

6.3.   side

6.3.1.    length of side

6.3.2.    relations among sides

6.4.   diagonal

6.5.   (line) segment

6.5.1.    relations among segments

6.5.1.1.     relations among segments (equalities)

6.5.1.2.     relations among segments (inequalities)

6.6.   perimeter

6.6.1.    length of perimeter

6.7.   area

6.7.1.    measure of area

6.7.1.1.     measure of the area of a quadrilateral(equalities)

6.7.1.2.     measure of the area of a quadrilateral(inequalities)

6.7.2.    measure of the area of generated figures

6.7.3.    ratio of areas

6.7.4.    relations among areas

6.8.   congruency of quadrilaterals

6.9.   special points

6.10.       concurrent lines

6.11.       lines which are: parallel, perpendicular, bisector, . . .

6.11.1.   the lines are parallel

6.11.2.   the lines are perpendicular

6.12.       other problems relating to this part

7.    Section 7.  n-gons (n≥5)

7.1.   definition and theorem

7.2.   angle

7.2.1.    measure of angle

7.2.2.    relations among angles

7.2.2.1.     relations among angles (equalities)

7.2.2.2.     relations among angles (inequalities)

7.3.   side

7.3.1.    number of sides

7.3.2.    relations among sides

7.3.2.1.     relations among sides (equalities)

7.3.2.2.     relations among segments (inequalities)

7.3.3.    other problems relating to this part

7.4.   diagonal

7.4.1.    length of diagonal

7.4.2.    number of diagonals

7.4.3.    other problems relating to this part

7.5.   (line) segment

7.5.1.    relations among segments

7.5.1.1.     relations among segments (equalities)

7.5.1.2.     relations among segments (inequalities)

7.6.   perimeter

7.6.1.    length of perimeter

7.6.2.    relations among perimeters

7.6.2.1.     relations among perimeters (equalities)

7.6.2.2.     relations among perimeters (inequalities)

7.7.   area

7.7.1.    measure of area

7.7.2.    relations among area

7.7.2.1.     relations among area (equalities)

7.7.3.    other problems relating to this part

7.8.       vertices

7.9.       sets of plane

7.10.       other problems relating to this part

Guidance and solution

References