A(x1,y1), B(x2,y2), C(x3,y3)
To calculate the coordinate of the circumcenter, we have to solve any 2 bisector equations of any of the three lines AB, BC, or AC, then find out the intersection points. i.e.
The midpoint of side AB = ⟮2x1+x2,2y1+y2⟯=(x12,y12)
And the slope of AB = x2−x1y2−y1=m12
∴ The slope of the perpendicular bisector of side AB = m12−1=m21
The equation of the perpendicular bisector of AB is therefore,
y−y12=m21(x−x12) −−−(i)
Similarly,
For side AC
The midpoint of side AC =
⟮2x1+x3,2y1+y3⟯=(x13,y13)
And its slope = x3−x1y3−y1=m13
Making the slope of its perpendicular bisector = m13−1=m31
The equation of the perpendicular bisector of length AC is therefore,
y−y13=m31(x−x13) −−−(ii)
On solving equations (i) and (ii), the values of x and y is the coordinate of the circumcenter
OR, using formula method;
Circumcenter(X,Y)=⟮sin2A+sin2B+sin2Cx1sin2A+x2sin2B+x3sin2C,sin2A+sin2B+sin2Cy1sin2A+y2sin2B+y3sin2C⟯
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