#"the points A,B,C are on the circle."#
#"we can write the fallowing equations"#
#A(5,2)#
#x^2+y^2+ax+bx+c=0#
#5^2+2^2+5a+2b+c=0#
#25+4+5a+2b+c=0#
#5a+2b+c=-29" (1)"#
#B(8,1)#
#x^2+y^2+ax+bx+c=0#
#8^2+1^2+8a+b+c=0#
#64+1+8a+b+c=0#
#8a+b+c=-65" (2)"#
#C(3,4)#
#x^2+y^2+ax+bx+c=0#
#3^2+4^2+3a+4b+c=0#
#9+16+3a+4b+c=0#
#3a+4b+c=-25" (3)"#
#"let's add the equation (2) to (3)"#
#11a+5b+2c=-90" (4)"#
#"expand the equation (1) by 2"#
#10a+4b+2c=-58" (5)"#
#"subtract the equation (5) from (4)"#
#a+b=-32" (5)"#
#"subtract the equation (1) from (2)"#
#3a-b=-36" (6)"#
#"now add (5) to (6)"#
#4a=-68#
#color(red)(a=-17)#
#"use (5)"#
#-17+b=-32#
#b=-32+17#
#color(red)(b=-15)#
#"use (2)"#
#8(-17)-15+c=-65#
#-136-15+c=-65#
#c=-65+151#
#color(red)(c=86)#
#"r:radius of circle"#
#r=sqrt(a^2+b^2-4c)/2#
#r=sqrt(289+225-344)/2#
#r=sqrt(514-344)#
#r=sqrt(170)/2#
#r^2=170/4=42.5#
#area=pi*r^2#
#area=42.5pi#