prove that the radius of the circum circle of a scalene triangle is abc/4area

Asked by  | 23rd Apr, 2008, 10:10: AM

Expert Answer:

The circumcircle always passes through all three vertices of a triangle. Its center is at the point where all the perpendicular bisectors of the triangle's sides meet.

  1. The radius of the circumcircle. The radius is given by the formula
    abc/sq root of ((a+b+c)(b+c-a)(c+a-b)(a+b-c))
    where a,b,c are the lengths of the sides of the triangle.
  2. Again we know area of a scalene triangle is sq root of s(s-a)(s-b)(s-c)  where s is the semi perimeter = (a+b+c)/2
  3. we can slove for area   by putting the value of s = (a+b+c)/2   and we will get area= sq root of ((a+b+c)(b+c-a)(c+a-b)(a+b-c)) by 4 That means 4 x area = sq root of ((a+b+c)(b+c-a)(c+a-b)(a+b-c))
  4. put this value in radius formulae ,so we will get radius = abc/4area

Answered by  | 25th Jul, 2008, 11:45: AM

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