Question

If you know the length (a,b,c) of the three sides of a triangle, the radius of its circumcircle is given by the formula: radius = a b c √ ( a + b + c ) ( b + c − a ) ( c + a − b ) ( a + b − c ) proof

Answer 1

Heya user,

Let the triangle be ABC.. and we make the Circumcircle...
--> We label the sides and the Angles as well as the center of circle.

Now, draw CX such that point X lies on Arc AB and CX passes through center of the circle...
Hence, ∠CXB = ∠CAB or ∠X = ∠A

Also, in ΔCXB, since CX is the diameter,
the ∠CBX = 90°...

Thus, sin X = BC / CX ----> [ where CX is the diameter ]
=> sin X = BC / 2R -->  [ where R is the circumradius ]
=> sin A = sin X = a / 2R
=> a / sin A = 2R;

Now, area of ΔABC = 1/2 * AC * AB sin A = 1/2 * bc sin A
=> area of ΔABC = 1/2 * b * c * a / 2R
=> area of ΔABC = 1/2 * abc /2R
=> [ ABC ] = abc / 4R ------> ( 1 )

Now, by heron's formula,
[ ABC ] = sqrt [ ( a + b + c ) ( a + b - c ) ( a + c - b ) ( b + c - a ) ] / 4 ---> (2)

Equating (1) & (2),

sqrt [ ( a + b + c ) ( a + b - c ) ( a + c - b ) ( b + c - a ) ] / 4 = abc / 4R
=> sqrt [ ( a + b + c ) ( a + b - c ) ( a + c - b ) ( b + c - a ) ]  = abc / R
=> R = abc / sqrt [ ( a + b + c ) ( a + b - c ) ( a + c - b ) ( b + c - a ) ]