The vertices of the triangle A (1; 3), B (-1; 1), C (2; 2). Find the coordinates of the center of the circle circumscribed about it.

1. Given:
A (1; 3);
AT 11);
C (2; 2).
2. Let O (x; y) be the center of the circumcircle. Then:
AO ^ 2 = (x – 1) ^ 2 + (y – 3) ^ 2;
BO ^ 2 = (x + 1) ^ 2 + (y – 1) ^ 2;
CO ^ 2 = (x – 2) ^ 2 + (y – 2) ^ 2;
AO ^ 2 = BO ^ 2 = CO ^ 2;
(x – 1) ^ 2 + (y – 3) ^ 2 = (x + 1) ^ 2 + (y – 1) ^ 2 = (x – 2) ^ 2 + (y – 2) ^ 2;
x ^ 2 – 2x + 1 + y ^ 2 – 6y + 9 = x ^ 2 + 2x + 1 + y ^ 2 – 2y + 1 = x ^ 2 – 4x + 4 + y ^ 2 – 4y + 4;
-2x + 1 – 6y + 9 = 2x + 1 – 2y + 1 = -4x + 4 – 4y + 4;
{-2x + 1 – 6y + 9 = 2x + 1 – 2y + 1;
{-2x + 1 – 6y + 9 = -4x + 4 – 4y + 4;
{-4x – 4y = -8;
{2x – 2y = -2;
{x + y = 2;
{x – y = -1;
{2x = 2 + (-1);
{2y = 2 – (-1);
{2x = 1;
{2y = 3;
{x = 1/2;
{y = 3/2.
Answer: (1/2; 3/2).