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+32, 4+42−3+32, 4+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)2, 4+(−3)23+(−4)2, 4+(−3)2)
Coordinates of N(−12, 12)−12, 12)
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 =−1−1m =−1
Perpendicular bisector of BC is a line passing through N (−12, 12−12, 12) 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