Essential Geometric Constructions

These notes cover essential geometric constructions that form the foundation of Discrete Geometry. By mastering these techniques, you can create precise shapes and understand their properties, enhancing your skills in both theoretical and practical applications of geometry.

1. Constructing a perpendicular bisector of a line segment

   • Use a compass to draw arcs from both endpoints of the segment.
   • The intersection points of the arcs determine the perpendicular bisector.
   • This construction ensures that the bisector divides the segment into two equal parts at a right angle.

2. Constructing an angle bisector

   • Place the compass point at the vertex of the angle and draw an arc that intersects both rays.
   • From the intersection points, draw arcs of equal radius to find the bisector.
   • The angle bisector divides the angle into two equal angles.

3. Constructing a line parallel to a given line through a point

   • Use a straightedge to draw a transversal line through the given point.
   • Measure the angle formed with the given line and replicate it on the other side of the transversal.
   • This ensures the new line is parallel to the original line.

4. Constructing a perpendicular line to a given line through a point

   • Place the compass point on the given line at the specified point and draw an arc that intersects the line.
   • From the intersection points, draw arcs above and below the line to find the perpendicular.
   • Connect the point to the intersection of the arcs to create a perpendicular line.

5. Constructing an equilateral triangle

   • Start with a line segment as one side of the triangle.
   • Use a compass to draw arcs from each endpoint of the segment with the same radius equal to the segment length.
   • The intersection of the arcs determines the third vertex, completing the triangle.

6. Constructing a square

   • Begin with a line segment for one side of the square.
   • Construct a perpendicular line at one endpoint and mark the same length as the segment.
   • Repeat for the other endpoint, then connect the vertices to form the square.

7. Constructing a regular hexagon

   • Draw a circle with a compass, using the radius as the side length.
   • Mark points on the circle by dividing it into six equal parts using the compass.
   • Connect the points to form the hexagon.

8. Constructing the circumcenter of a triangle

   • Construct the perpendicular bisectors of at least two sides of the triangle.
   • The intersection of these bisectors is the circumcenter, equidistant from all triangle vertices.
   • This point is the center of the circumcircle that passes through all three vertices.

9. Constructing the incenter of a triangle

   • Construct the angle bisectors of at least two angles of the triangle.
   • The intersection of these bisectors is the incenter, which is equidistant from all sides of the triangle.
   • This point is the center of the incircle that touches all three sides.

10. Constructing a tangent line to a circle from an external point

    • Draw a line from the external point to the center of the circle.
    • Construct a perpendicular line from the center to the circle's edge.
    • The tangent line touches the circle at exactly one point, forming a right angle with the radius at that point.