Centroid

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

In the adjoining figure, G is the centroid.k11

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.k21

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

k31The 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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Orthocenterk41

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 Trianglek51

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, thenk61

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.

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