Centroid, circumcentre, orthocentre, incentre of triangle

Centroid:

• The centroid of a triangle is the point of intersection of medians. It divides medians in 2 : 1 ratio.
• If  A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) are vertices of triangle ABC, then coordinates of centroid is
  G = [(x₁ + x₂ + x₃)/ 3, (y₁ + y₂ + y₃)/ 3]
  
  

In center: 

• Point of intersection of angular bisectors
• Coordinates of
  I = [(ax₁ + bx₂ + cx₃)/ a + b + c, (ay₁ + by₂ + cy₃)/ a + b + c]
• Where a, b, c are sides of triangle ABC.

Circumcentre:

• Point of intersection of perpendicular bisectors.
• Co-ordinates of circumcentre O is
  O = [(x₁sin2A+x₂sin2B+x₃sin2c)/ sin2A+sin2B+sin2c, (y₁sin2A+y₂sin2B+y₃sin2c)/ sin2A+sin2B+sin2c]

Orthocenter:

• Point of intersection of altitudes of triangle ABC.
• Coordinates of orthocenter H is
  H = [(x₁tanA + x₂tanB + x₃tanc)/ tanA+tanB+tanc, (y₁tanA + y₂tanB + y₃tanc)/ tanA+tanB+tanc]

Note:

• Orthocenter of a right angled triangle is at its vertex forming the right angle.
• The orthocenter H, circumcentre O and centroid G of a triangle are collinear and G divides H, O in ratio 2 : 1 i.e., HG : OG = 2: 1.
• Circumcentre of a right angled triangle is mid-point of hypotenuse.

Nine Point Circle:

• Let ABC be triangle such that AD, BE and CF are its altitudes, H, I, J are midpoints of line segments of sides BC, CA, AB respectively; K, L, M are midpoints of joining orthocenter (O) to angular points A, B, C. 
• These 9 points (D, E, F, H, I, J, K, L, M) are concyclic and the circle passing through nine points is nine point circle and Center is known as nine- point center.