Geometry Lecture Notes: Incenter, Circumcenter, Centroid, and Orthocenter

Incenter

• Definition: The point where the three angle bisectors of a triangle intersect.
• Properties:
  • The incenter is the center of a circle that can be inscribed inside the triangle.
  • This circle is tangent to the sides of the triangle.
  • Radii from the incenter to the tangent points are congruent.

Circumcenter

• Definition: The point where the three perpendicular bisectors of a triangle's sides intersect.
• Properties:
  • The circumcenter is the center of a circle that can circumscribe the triangle.
  • All radii from the circumcenter to the vertices of the triangle are congruent.
  • Position:
    • Right Triangle: The circumcenter lies on the triangle.
    • Obtuse Triangle: The circumcenter lies outside the triangle.
    • Acute Triangle: The circumcenter lies inside the triangle.
• Construction Note: Perpendicular bisectors may not pass through the opposite vertex.

Centroid

• Definition: The point where the three medians of a triangle intersect.
• Properties:
  • The centroid divides each median into two segments, with a 2:1 ratio.
  • Acts as the "center of mass" or balancing point of the triangle.
• Segment Division:
  • Portion from vertex to centroid is 2/3 of the median.
  • Portion from side to centroid is 1/3 of the median.

Orthocenter

• Definition: The point where the three altitudes of a triangle intersect.
• Properties:
  • Right Triangle: Orthocenter lies on the triangle.
  • Acute Triangle: Orthocenter lies inside the triangle.
  • Obtuse Triangle: Orthocenter lies outside the triangle.
  • Altitudes are measured perpendicular from a vertex to the opposite side.

Additional Notes

• For constructions using a compass and straightedge, refer to additional resources or videos.