If the sides of a triangle are #a#, #b# and #c#, then the area of the triangle #Delta# is given by the formula
#Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#
and radius of circumscribed circle is #(abc)/(4Delta)#
Hence let us find the sides of triangle formed by #(6,8)#, #(1,2)# and #(3,9)#. This will be surely distance between pair of points, which is
#a=sqrt((1-6)^2+(2-8)^2)=sqrt(25+36)=sqrt61=7.810#
#b=sqrt((3-1)^2+(9-2)^2)=sqrt(4+49)=sqrt53=7.280# and
#c=sqrt((3-6)^2+(9-8)^2)=sqrt(9+1)=sqrt10=3.162#
Hence #s=1/2(7.810+7.280+3.162)=1/2xx18.252=9.126#
and #Delta=sqrt(9.126xx(9.126-7.810)xx(9.126-7.280)xx(9.126-3.162)#
= #sqrt(9.126xx1.316xx1.846xx5.964)=sqrt132.2226=11.499#
And radius of circumscribed circle is
#(7.810xx7.280xx3.162)/(4xx11.499)=3.909#
And area of circumscribed circle is #3.1416xx(3.909)^2=48.005#
