To calculate the area of the circle, we must calculate the radius #r# of the circle
Let the center of the circle be #O=(a,b)#
Then,
#(5-a)^2+(1-b)^2=r^2#.......#(1)#
#(3-a)^2+(9-b)^2=r^2#..........#(2)#
#(4-a)^2+(7-b)^2=r^2#.........#(3)#
We have #3# equations with #3# unknowns
From #(1)# and #(2)#, we get
#25-10a+a^2+1-2b+b^2=9-6a+a^2+81-18b+b^2#
#4a-16b=-54#
#2a-8b=-27#.............#(4)#
From #(2)# and #(3)#, we get
#9-6a+a^2+81-18b+b^2=16-8a+a^2+49-14b+b^2#
#2a-4b=-25#
#2a-4b=-25#..............#(5)#
From equations #(4)# and #(5)#, we get
#-27+8b-4b=-25#
#4b=2#
#b=2/4=1/2#
#2a=-25+4b=-25+2=-23#, #=>#, #a=-23/2#
The center of the circle is #=(-23/2,1/2)#
#r^2=(5-a)^2+(1-b)^2=(5+23/2)^2+(1-1/2)^2#
#=1089/4+1/4#
#=1090/4#
The area of the circle is
#A=pi*r^2=1089/4*pi=855.3#