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