The points of intersections of certain lines related to a triangle are defined below.
These are: a) Centroid, b) Circumcentre, c) Orthocentre, d) Incentre and e) Ex-centres.
The formulae for them or method of solving in co-ordinate geometry is explained.
Let A(x1, y1), B(x2, y2) and C(x3, y3) be the vertices of a triangle ABC.
a) Centroid (G)
The medians of a triangle are concurrent, that is, they intersect at the same point.
The point of intersection of the medians is called the centroid of a triangle.
It is denoted by G, which always lies inside the triangle.
b) Circumcentre (O)
The right bisectors of the sides of a triangle are concurrent, that is, they intersect at the same point.
The point of intersection of the right bisectors of the sides of a triangle is called its circumcenter.
It is denoted by O.
Circumradius, R = OA = OB = OC.
Steps to find circumcentre
(i) When vertices of a triangle are given
Let 'O' be the circumcentre of a triangle formed by the vertices A, B, C.
Step-1: Find the following two equations: OA = OB and OB = OC
Step-2: Find the point of intersection of the two equations above.
(ii) When the equations of sides are given
Find the vertices of a triangle by solving pairs of equations and proceed as in (i) above.