Centroid

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

In the adjoining figure, $G$ is the centroid.  $AP, BQ \ and \ CR$  are the medians which divide the corresponding sides $BC, AC \ and \ AB$  respectively in two equal halves.

Hence, $BP=PC, CQ=QA \ and \ AR=RB$

The centroid of the triangle always divide each of the medians in the ratio of $2:1$

Therefore, $AG:GP = 2:1,$ $BG:GQ=2:1 \ and \ CG:GR=2:1$ $\\$

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 $IF=IE=ID$

If you draw a circle, with $I$ as the center, then the radius of this Incircle would be $IF \ or \ IE \ or \ ID$

In the adjoining figure, $AI, BI \ and \ CI$  are bisectors of angles $A, B \ and \ C$  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 $BD=CD, CE=AE \ and \ AF=CF$.

The distance from the center to the three vertices $A, B \ and \ C$  are equal. i.e. $AO = BO=CO$

If you draw a circle with $O$  as the center, and the radius $AO \ or \ BO \ or \ CO$, 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, $AD \perp BC, AD \perp BE \ and \ CF \perp AB$ $\\$

Properties of Isosceles Triangle If a triangle is an Isosceles triangle, then

Median $AD = bisector of \angle A$

= perpendicular bisector of opposite side $BC$

= Altitude of corresponding side $BC$ $\\$

Properties of Equilateral Triangle

If the triangle is an equilateral triangle, then Median $AD =$  bisector of $\angle A$

= perpendicular bisector of opposite side $BC$

= Altitude of corresponding side $BC$

Median $BE =$  bisector of $\angle B$

= perpendicular bisector of opposite side $AC$

= Altitude of corresponding side $AC$

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