Some basic definitions:

Centroid (G): The centroid of a triangle divides each median in ratio of 2:1(2 from vertex and 1 from opposite side). The coordinates of centroid can be given by

                    .

Incentre (I): The coordinates of the in-centre of a triangle having vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) can be given by   where a, b and c are length of sides BC, CA and AB respectively.  

Orthocentre (H): The co-ordinates of orthocentre of triangle A(x₁, y₁), B(x₂, y₂), C(x₃,y₃) are.

Circumcentre (C):

• The coordinates of circum centre of the triangle having vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) is given by

        .

• Note: The circum centre of the right angled triangle is midpoint of its hypotenuse.     

• In the equilateral triangle centroid, in centre, orthocentre and circum centre all the coincides.

• The centroid divides line joining the orthocentre and circum centre in the ratio of 2:1 internally.

Illustration: If the vertices of a triangle be (2√3, 6),  (3, -√3) and (0, 0). Then find its orthocentre and circumcentre.

Key concept: In a right angled triangle orthocentre is the right angled vertex, and circumcentre is the mid point of the hypotenuse. 

Solution: Clearly, the given triangle is right angled at the vertex(0, 0), so its orthocentre is  (0, 0) and circumcentre will be the mid point of hypotenuse which is   .

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