Centroid, Incentre, Circumcentre & Orthocenter

Centroid:

Definition:

  • Coordinates: The centroid of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is given by: (x1+x2+x33,y1+y2+y33)
  • Concept: The centroid is the point of intersection of the medians of a triangle.
  • Property: Divides each median into segments with a ratio of 2:1.

Incenter:

Definition:

  • Coordinates: The incenter of a triangle with vertices (x1,y1), (x2,y2) and (x3,y3) is given by: (ax1+bx2+cx3a+b+c,ay1+by2+cy3a+b+c) where a,b, and c are the side lengths opposite the respective vertices.
  • Concept: The incenter is the point of intersection of the angle bisectors of a triangle.
  • Property: Equidistant from the sides of the triangle.

Circumcenter:

Definition:

  • Coordinates: The circumcenter of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is given by: (D12K,D22K) where D1 and D2 are certain determinants and K is the area of the triangle.
  • Concept: The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle.
  • Property: Equidistant from the vertices of the triangle.

Orthocenter:

Definition:

  • Coordinates: The orthocenter of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is given by: (i=13xi(hi2ai2)i=13(hi2ai2),i=13yi(hi2ai2)i=13(hi2ai2)) where hi is the length of the altitude and ai is the length of the side opposite the ith vertex.
  • Concept: The orthocenter is the point of intersection of the altitudes of a triangle.
  • Property: May lie inside, outside, or on the triangle.

Relationships:

  • Equilateral Triangle: For an equilateral triangle, all these points coincide at a single point.
  • Different Types of Triangles: The relative positions of these points may vary for different types of triangles (acute, obtuse, or right-angled).