# Centroid of a Triangle

**The point at which the medians of a triangle intersect is called the centroid of the triangle.**

Let ABC be a given triangle with vertices A(
Since D is the mid-point of BC, its coordinates are [(

*x*_{1},*y*_{1}), B(*x*_{2},*y*_{2}) and C(*x*_{3},*y*_{3}).*x*_{2}_{ }+*x*_{3}_{})/2, (*y*_{2}_{ }+*y*_{3}_{})/2]Let G(

*x*,*y*) be a point dividing AD in the ratio 2:1.Then,

and

Similarly, the coordinates of a point which divides BE in the ratio 2:1 as well as those of the point which divides CF in the ratio 2:1 are

# Incentre of a Triangle

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The point at which the bisectors of the angles of a triangle intersect, is called the in-centre of the triangle. From geometry, we know that the bisector of an angle of a triangle divides the opposite side in the ratio of length of remaining sides. Hence, the bisectors of the angle of âˆ†ABC are concurrent, and meet at a point, called In-centre.

# Area of Triangle

Let ABC be a given triangle whose vertices are A(*x*

_{1},

*y*

_{1}), B(

*x*

_{2},

*y*

_{2}) and C(

*x*

_{3},

*y*

_{3}).

Area of the triangle

=

If we interchange the order of any two vertices of the DABC, we obtain a negative value of the area. However, the area shall always be taken to be positive.