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Centroid of a Triangle

The point at which the medians of a triangle intersect is called the centroid of the triangle.
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Let ABC be a given triangle with vertices A(x1y1), B(x2y2) and C(x3y3).
Since D is the mid-point of BC, its coordinates are [(x2 x3)/2, (y2 y3)/2]
Let G(xy) be a point dividing AD in the ratio 2:1.
Then, Description: 14118.png
and Description: 14132.png
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
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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 ∆ABC are concurrent, and meet at a point, called In-centre.
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Area of Triangle

Let ABC be a given triangle whose vertices are A(x1y1), B(x2y2) and C(x3y3).
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Area of the triangle

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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.

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