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,
#(1-a)^2+(5-b)^2=r^2#.......#(1)#
#(7-a)^2+(9-b)^2=r^2#..........#(2)#
#(4-a)^2+(2-b)^2=r^2#.........#(3)#
We have #3# equations with #3# unknowns
From #(1)# and #(2)#, we get
#1-2a+a^2+25-10b+b^2=49-14a+a^2+81-18b+b^2#
#12a+8b=130-26=104#
#3a+2b=26#.............#(4)#
From #(3)# and #(2)#, we get
#16-8a+a^2+4-4b+b^2=49-14a+a^2+81-18b+b^2#
#6a+14b=130-20=110#
#3a+7b=55#..............#(5)#
From equations #(4)# and #(5)#, we get
#26-2b=55-7b#, #=>#, #5b=29#, #b=29/5#
#3a=26-2b=26-58/5=72/5#, #=>#, #a=24/5#
The center of the circle is #=(24/5,29/5)#
#r^2=(1-24/5)^2+(5-29/5)^2=(-19/5)^2+(-4/5)^2#
#=361/25+16/25#
#=377/25#
The area of the circle is
#A=pi*r^2=pi*377/25=47.38u^2#