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 #(1,3)#, #(2,5)# and #(6,4)#. This will be surely distance between pair of points, which is
#a=sqrt((2-1)^2+(5-3)^2)=sqrt(1+4)=sqrt5=2.2361#
#b=sqrt((6-2)^2+(4-5)^2)=sqrt(16+1)=sqrt17=4.1231# and
#c=sqrt((6-1)^2+(4-3)^2)=sqrt(25+1)=sqrt26=5.0990#
Hence #s=1/2(2.2361+4.1231+5.0990)=1/2xx11.4582=5.7291#
and #Delta=sqrt(5.7291xx(5.7291-2.2361)xx(5.7291-4.1231)xx(5.7291-5.0990)#
= #sqrt(5.7291xx3.4930xx1.6060xx0.6301)=sqrt20.2507=4.5001#
And radius of circumscribed circle is
#(2.2361xx4.1231xx5.0990)/(4xx4.5001)=2.6117#
And area of circumscribed circle is #3.1416xx(2.6117)^2=21.4288#
