What are the coordinates of the circumcenter of a triangle with vertices a(1,5)a(1,5), b(3,3)b(3,3), and c(3,5)c(3,5)?

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