Circumcenter of a triangle is the intersection point of the perpendicular bisectors of the three sides.

Perpendicular bisector of a line segment  is a line passing through the mid-point of the line segment and is perpendicular to it.

Perpendicular bisector of AB:
Mid-point of AB, M(−3+324+42−3+324+42)

Coordinates of M(0, 4)0, 4)

Gradient or slope of AB, m = 4−4−3−3 =04−4−3−3 =0

Gradient or slope of line perpendicular to AB = 1m =∞1m =∞

⇒⇒ Perpendicular line to AB is a vertical line on xy plane.

Perpendicular bisector of AB is a vertical line passing through M(0,4). It’s equation: x=0   ———-Line 1

Perpendicular bisector of BC:
Mid-point of BC, N(3+(−4)24+(−3)23+(−4)24+(−3)2)

Coordinates of  N(1212)1212)

Gradient or slope of BC, m = −3−4−4−3 =1−3−4−4−3 =1

Gradient or slope of line perpendicular to BC = 1m =−11m =−1

Perpendicular bisector of BC is a line passing through N (12121212) and is having a slope -1.

Equation of perpendicular bisector of BC:

y−12 =−1(x−(−12))y−12 =−1(x−(−12))

y−12 =−x−12y−12 =−x−12

y = x  ——— Line 2

Circumcenter is point of intersection of Line 1 and Line 2.

x= 0, y =0