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If sides of a triangle 3,4,5 cm .Find the distance between incentre and circumcentre.

 Nov 11, 2019
 #1
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Triangle
Sides: a = 3 b = 4 c = 5

Area: T = 6
Perimeter: p = 12
Semiperimeter: s = 6
Inradius: r = 1
Circumradius: R = 2.5
The distance between the incenter and the circumcenter is sqrt(R(R-2r)), where R is the circumradius and r is the inradius, a result known as the Euler triangle formula.
The Distance=Sqrt(2.5*(2.5 - 2*1))
                        =Sqrt(2.5*(0.5))
                        =Sqrt(1.25)
                        =1.118 cm - distance between incenter and circumcenter.

 Nov 11, 2019
 #2
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Let  A  = (0, 0)

Let B  =  (0, 3)

Let C  =  (4, 0)

 

We can find the incenter  thusly  :

 

[ Length of side opposite A * xcoordinate of A + Length of side opposite B * x coordinate of B + Length of side opposite C*xcoordinate of C ] ,  [ Length of side opposite A * ycoordinate of A + Length of side opposite B * y coordinate of B + Length of side opposite C*ycoordinate of C ]

 

 ( [(5(0)  + 0 (4) + 4(3)] / 12  , [5(0)  + 3(4)  + 0(3) ] / 12  )  =

 

(1 , 1 )

 

In a right triangle.....the circumcenter is located at the midpoint of the hypotenuse....this point is

 

(2, 3/2)

 

So....the distance between the incenter and the circumcenter is

 

sqrt  [ ( 2 - 1)^2  + (3/2 - 1)^2  ]  =

 

sqrt [ 1*2  +  (1/2)^2  ] =

 

sqrt [ 1 + 1/4  ]  =

 

sqrt  [ 5/4]   =   sqrt  (5)  / 2

 

 

cool cool cool

 Nov 11, 2019

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