Question

find the center of the circle passing through {5,-8} , {2,-9} , {2,1}

Answer 1

Let the vertices of the given triangle be
A (5, -8)
B (2, -9)
C (2, 1)

Let the midpoints of each side (AB, BC & CA) be Q, R, P  respectively.
The place where the medians intersect is the centroid, which is the centre of the circle circumscribing the given triangle.

By using the section formula 
$x = \frac{m x_{2}+ nx_{1} }{m+n} , y = \frac{m y_{2}+ ny_{1} }{m+n}$

(use m= 1 = n)

We get coordinates
P ( 7/2, -7/2)
Q (7/2, -17/2)
R (2, -4)

It is a property of triangle that the centroid divides each median in the ratio 2:1

This implies 
AO:OR = 2:1
m = 2
n = 1

By using the section formula
$x = \frac{2(2)+ 1(5) }{2+1} \\ x = \frac{9}{3} \\ x= 3 \\ \\ y= \frac{2(-4)+ 1(-8) }{2+1} \\ y = \frac{-16}{3} Hence, coordinates of O are (3, -16)$