Geo2.44

Find the circumcenter of the triangle formed by the points $(2,1),(1,-2),(-2,3)\,$

Let the points be $A,B,C,D\,$ respectively.

Midpoint of $BC=D=\left(\frac{-1}{2},\frac{1}{2}\right)\,$

Midpoint of $AB=F=\left(\frac{3}{2},\frac{-1}{2}\right)\,$

Slope of $BC=\frac{-2-3}{1+2}=-\frac{5}{3}\,$

Let S be the circum center.Therefore,the slope of SD is $\frac{3}{5}\,$

Equation of the perpendicular bisector SD of BC is $y-\frac{1}{2}=\frac{3}{5}(x+\frac{1}{2}),6x-10y+8=0\,$

Slope of $AB=3\,$ . Slope of $SF=\frac{-1}{3}\,$

Equation of the perpendicular bisector SF of AB is $y+\frac{1}{2}=\frac{-1}{3}(x-\frac{3}{2}),2x+6y=0,x+3y=0\,$

Solving these two equations,we get the circumcenter as $\left(-\frac{6}{7},\frac{2}{7}\right)\,$