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

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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.

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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.

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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,  $AD \perp BC, AD \perp BE \ and \ CF \perp AB$

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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  $BC$

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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.