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 #(4,4)#, #(8,9)# and #(3,1)#. This will be surely distance between pair of points, which is
#a=sqrt((8-4)^2+(9-4)^2)=sqrt(16+25)=sqrt41=6.4031#
#b=sqrt((3-8)^2+(1-9)^2)=sqrt(25+64)=sqrt89=9.4340# and
#c=sqrt((3-4)^2+(1-4)^2)=sqrt(1+9)=sqrt10=3.1623#
Hence #s=1/2(6.4031+9.4340+3.1623)=1/2xx18.9994=9.4997#
and #Delta=sqrt(9.4997xx(9.4997-6.4031)xx(9.4997-9.4340)xx(9.4997-3.1623)#
= #sqrt(9.4997xx3.0966xx0.0657xx6.3374)=sqrt12.2482=3.4997#
And radius of circumscribed circle is
#(6.4031xx9.4340xx3.1623)/(4xx3.4997)=13.6458#
And area of circumscribed circle is #3.1416xx(13.6458)^2=584.99#
