Centroid of Triangle

The centroid of a triangle is the point of concurrency of the medians. The centroid G of the triangle ABC, divides the median AD, in the ratio of 2 : 1.
 
Illustration:
 
Find the centroid of the triangle the coordinates of whose vertices are given by A(x1, y1), B(x2, y2) and C(x3, y3) respectively.
 
 

2475_Centroid incentre and circum centre 1.JPG

 
Solution:
 
             AG/AD = 2/1
 
 Since D is the midpoint of BC, coordinates of D are (x2+x3/2, y2+y3/2)
 
 Using the section formula, the coordinates of G are
 
             (2(x2+x3/2)+1.x1/2+1, 2(y2+y3/2)+1.y1/2+1)
 
         ⇒ Coordinates of G are (x1+x2+x3/3, y1+y2+y3/3).
 
 
Incentre of Triangle
 
The incentre ‘I’ of a triangle is the point of concurrency of the bisectors of the angles of the triangle.
 
Illustration:
 
Find the incentre of the triangle the coordinates of whose vertices are given by A(x1, y1), B(x2, y2), C(x3, y3).
 1022_Centroid incentre and circum centre 2.JPG
Solution:
 
By geometry, we know that BD/DC = AB/AC (since AD bisects ÐA).
 
If the lengths of the sides AB, BC and AC are c, a and b respectively, then BD/DC = AB/AC = c/b.
 
Coordinates of D are (bx2+cx3/b+c, by2+cy3/b+c)
 
IB bisects ÐB. Hence ID/IA = BD/BA = (ac/b+c)/c = a/c+b.
                                                     –  –
Let the coordinates of I be (x, y).
                                                        –
Then x = ax1+bx2+cx3/a+b+c, y = ay1+by2+cy3/a+b+c.
  

Circum Centre of Triangle 

 
This the point of concurrency of the perpendicular bisectors of the sides of the triangle. This is also the centre of the circle, passing through the vertices of the given triangle.
  

Orthocentre of Triangle

 
This is the point of concurrency of the altitudes of the triangle.
  
Excentre
 
Excentre of a triangle is the point of concurrency of bisectors of two exterior and third interior angle. Hence there are three excentres I1, I2 and I3 opposite to three vertices of a triangle.
  
If A(x1, y1), B(x2, y2) and C(x3, y3) are the vertices of a triangle ABC,
 2125_Centroid incentre and circum centre 3.JPG
coordinates of centre of ex-circle opposite to vertex A are given as
 
             I1(x, y) = (–ax1+bx2+cx3/a+b+c/–a+b+c, –ay1+by2+cy3/–a+b+c).
 
Similarly co-ordinates of centre of I2(x, y) and I3(x, y) are
 
             I2(x, y) = (ax1–bx2+cx3/a–b+c, ay1–by2+cy3/a–b+c)
 

             I3(x, y) = (ax1+bx2–cx3/a+b–c, ay1+by2–cy3/a+b–c)

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