Circumcircle of Triangle

What is the circumcenter of triangle \(ABC\) with vertices

\[A=(1, 4), B=(-2, 3), C=(5, 2)?\]

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A circumcenter, by definition, is the center of the circle in which a triangle is inscribed, For this problem, let \(O=(a, b)\) be the circumcenter of \(\triangle ABC.\) Then since the distances to \(O\) from the vertices are all equal, we have \[\lvert \overline{AO} \rvert=\lvert \overline{BO} \rvert=\lvert \overline{CO} \rvert.\] From the first equality, we have \[\begin{align} \lvert \overline{AO} \rvert^2&=\lvert \overline{BO} \rvert^2\\ (a-1)^2+(b-4)^2&=(a+2)^2+(b-3)^2\\ -2a+1-8b+16&=4a+4-6b+9\\ 3a+b&=2. \qquad (1) \end{align}\] Similarly, from the second equality, we have \[\begin{align} \lvert \overline{BO} \rvert^2&=\lvert \overline{CO} \rvert^2\\ (a+2)^2+(b-3)^2&=(a-5)^2+(b-2)^2\\ 4a+4-6b+9&=-10a+25-4b+4\\ 7a-b&=8. \qquad (2) \end{align}\] Taking \((1)+(2)\) gives \(a=1,\) which in turn gives \(b=-1.\)

Therefore, the circumcenter of triangle \(ABC\) is \(O=(1, -1).\) \( _\square \)