Geometry

1. Plane Geometry
   1. Basic Concepts of Plane Geometry
      1. Definition of Plane
         1. Understanding the flat two-dimensional surface
         2. Infinite extension and zero thickness
         3. Difference between a line and a plane
      2. Axial Symmetry
         1. Definition and properties
         2. Line of symmetry
         3. Reflective symmetry in plane shapes
      3. Coordinate Plane
         1. Cartesian coordinate system
         2. Plotting points using (x, y) coordinates
         3. Quadrants of the coordinate plane
   2. Two-dimensional Shapes
      1. Triangles
         1. Types of Triangles
            1. Equilateral Triangle: all sides and angles are equal
            2. Isosceles Triangle: two sides are equal, two angles are equal
            3. Scalene Triangle: all sides and angles are different
         2. Properties of Triangles
            1. Sum of internal angles equals 180 degrees
            2. The exterior angle theorem
         3. Congruence and Similarity
            1. SSS, SAS, ASA, AAS, and RHS congruence criteria
            2. AA, SAS, and SSS similarity criteria
         4. Advanced Theorems
            1. Pythagorean theorem: a² + b² = c² in right triangles
            2. Triangle inequality theorem
         5. Formulas
            1. Heron's formula for area
            2. Perimeter calculations
   3. Quadrilaterals
      1. Types and Properties
         1. Squares: equal sides and angles, diagonals bisect at right angles
         2. Rectangles: opposite sides equal, diagonals equal
         3. Parallelograms: opposite sides parallel and equal, diagonals bisect each other
         4. Rhombus: all sides equal, opposite angles equal, diagonals bisect at right angles
         5. Trapezoid (US) / Trapezium (UK): only one pair of parallel sides
      2. Properties
         1. Internal angles summing to 360 degrees
         2. Mid-segment theorem for trapezoids
      3. Formulas
         1. Area and perimeter for each type of quadrilateral
         2. Special properties of diagonals
   4. Circles
      1. Basic Components
         1. Radius, diameter, and relation between them
         2. Circumference and area formulas
      2. Circle Theorems
         1. Inscribed angle theorem
         2. Tangent and radius properties
         3. Chord properties, including perpendicular bisector
      3. Arc and Sector
         1. Defining arcs and central angles
         2. Length of an arc
         3. Area of a sector
   5. Polygons
      1. Definition and Classification
         1. Regular vs. irregular polygons
         2. Convex vs. concave polygons
      2. Properties
         1. Interior and exterior angle sum formulas
         2. Diagonal calculations
      3. Formulas
         1. Area of regular polygon based on side length and number of sides
         2. Perimeter calculations
   6. Special Plane Figures
      1. Constructing Loci
         1. Concept of a locus of points
         2. Loci generated by movement constraints
      2. Tessellations and Tilings
         1. Definition and examples of tessellations
         2. Regular and semi-regular tessellations
         3. Applications in art and design
      3. Conic Sections
         1. Definition and properties of ellipses, parabolas, and hyperbolas
         2. Standard equations and their characteristics
         3. Applications of conic sections in real-world contexts
   7. Advanced Topics in Plane Geometry
      1. Coordinate-Based Problem Solving
         1. Using coordinate geometry to solve plane geometry problems
         2. Equation of a line in the plane
         3. Finding the distance between points
      2. Vector Geometry
         1. Introduction to vectors in a plane
         2. Dot product and cross product in two dimensions
         3. Applications of vectors in plane geometry
      3. Analytical Geometry
         1. Study of geometric properties without visualization
         2. Coordinate, position, and equations-based exploration
   8. Applications of Plane Geometry
      1. Art and Design
         1. Exploration of geometric shapes and symmetry in art
         2. Use of plane geometry in architectural design plans
      2. Engineering
         1. Structural analysis using two-dimensional geometry
         2. The role of plane geometry in mechanical drawings
      3. Physics and Sciences
         1. Optics: understanding light paths and lenses
         2. Navigation: use of geometric principles in mapping and GPS
   9. Problem Solving in Plane Geometry
      1. Geometric Proofs
         1. Writing formal proofs and justifying steps in plane geometry
         2. Different methods: direct, contradiction, and coordinate
      2. Construction Techniques
         1. Practical use of compass and straightedge in constructions
         2. Classic problems: bisecting angles, constructing parallel lines
      3. Olympiad-style Problems
         1. Exploring complex and interesting geometry problems
         2. Logical and spatial reasoning development through practice