Centroid

The point of intersection of the three medians is called the centroid of the triangle.

In the adjoining figure, is the centroid.

are the medians which divide the corresponding sides respectively in two equal halves.

Hence,

The centroid of the triangle always divide each of the medians in the ratio of

Therefore,

Incentre

The point of intersection of the bisectors of the internal angles of a triangle is called the Incentre of the triangle.

The Incentre of the triangle is equidistant from each of the sides of the triangle.

Hence

If you draw a circle, with as the center, then the radius of this Incircle would be

In the adjoining figure, are bisectors of angles respectively.

Circumcenter

The point of intersection of the perpendicular bisectors of the three sides is the circumcenter of the triangle. In the adjoining diagram, you can see that .

The distance from the center to the three vertices are equal. i.e.

If you draw a circle with as the center, and the radius , the circle will encircle the triangle and touch the three vertices.

Orthocenter

The point where the three perpendiculars drawn from the vertices of a triangle to the opposite side of the triangles meet is called the orthocenter of the triangle.

In the adjoining figure,

Properties of Isosceles Triangle

If a triangle is an Isosceles triangle, then

Median

= perpendicular bisector of opposite side

= Altitude of corresponding side

Properties of Equilateral Triangle

If the triangle is an equilateral triangle, then

Median bisector of

= perpendicular bisector of opposite side

= Altitude of corresponding side

Median bisector of

= perpendicular bisector of opposite side

= Altitude of corresponding side

Also if is the centroid of the triangle, it is also the Incentre, it is also the circumcenter and also the orthocenter.