Geometry

1. Geometric Constructions
   1. Tools for Constructions
      1. Compass
         1. Types of compasses
         2. Using a compass to draw circles and arcs
      2. Straightedge
         1. Definition and types
         2. Basic uses in construction
      3. Ruler and Protractor
         1. Uses and limitations in geometric constructions
      4. Additional Tools
         1. Constructing with string or paper folding
         2. Digital tools for geometric construction
   2. Basic Construction Techniques
      1. Constructing a Perpendicular
         1. Perpendicular from a point on a line
         2. Perpendicular from a point off the line
      2. Bisecting a Line Segment
         1. Step-by-step process
         2. Real-world applications
      3. Constructing an Angle Bisector
         1. Procedure for bisecting a given angle
         2. Application in real-life scenarios
   3. Classical Problems and Constructions
      1. Constructing Triangles
         1. Given three sides (SSS construction)
         2. Given two angles and a side (ASA construction)
         3. Constructing given two sides and the included angle (SAS construction)
      2. Constructing Regular Polygons
         1. Regular hexagon and equilateral triangle using a compass and straightedge
         2. Techniques for other regular polygons
      3. Inscribing and Circumscribing Polygons
         1. Constructing an inscribed circle in a triangle
         2. Constructing a circumscribed circle around a triangle
   4. Advanced Constructions
      1. Dividing a segment into equal parts
         1. Using a compass and straightedge method
         2. Applications in design and architecture
      2. Constructing a Tangent to a Circle
         1. Tangent from a point on the circle
         2. Tangent from a point outside the circle
      3. Constructing a Parallelogram
         1. Given specific side lengths and angles
         2. Exploring alternative construction methods
   5. Historical Constructions
      1. Ancient Greek Constructions
         1. Euclidean methods and notable constructions
         2. Limitations of early geometric tools
      2. Constructible Numbers and Their Properties
         1. Algebraic implications in constructions
         2. Historical theories and their impact on geometry
   6. Impossible Constructions
      1. The Trisecting an Angle Problem
         1. Using classic tools: why it's impossible
         2. Non-classical solutions
      2. Doubling the Cube
         1. The Delian problem explained
         2. Historical attempts and implications
      3. Squaring the Circle
         1. Pi and the challenge of perfect squares
         2. The mathematical reasoning behind impossibility
   7. Application of Constructions
      1. Art and Design
         1. Geometric approaches in art techniques
         2. Use in pattern creation and symmetry
      2. Education
         1. Teaching fundamental geometry through constructions
         2. Enhancing spatial visualization skills
      3. Practical Uses in Daily Life
         1. Layout planning and architectural designs
         2. Problem-solving in practical circumstances