The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle meet. It is equidistant from the vertices of the triangle.

Steps to Find the Circumcenter:

1. Identify the vertices of the triangle:
   $A(0, 4)$, $B(5, 0)$, $C(-2, 3)$.

2. Calculate the midpoints of two sides of the triangle:

   • Midpoint of $AB$: $\text{Midpoint of } AB = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{0 + 5}{2}, \frac{4 + 0}{2}\right) = (2.5, 2).$
   • Midpoint of $AC$: $\text{Midpoint of } AC = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{0 + (-2)}{2}, \frac{4 + 3}{2}\right) = (-1, 3.5).$

3. Find the slopes of the sides:

   • Slope of $AB$: $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 4}{5 - 0} = -\frac{4}{5}.$
   • Slope of $AC$: $m_{AC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 4}{-2 - 0} = \frac{-1}{-2} = \frac{1}{2}.$

4. Calculate the perpendicular slopes:

   • Perpendicular slope to $AB$: $m_{\perp AB} = -\frac{1}{m_{AB}} = \frac{5}{4}.$
   • Perpendicular slope to $AC$: $m_{\perp AC} = -\frac{1}{m_{AC}} = -2.$

5. Write the equations of the perpendicular bisectors:

   • Perpendicular bisector of $AB$ (passing through $(2.5, 2)$): $y - 2 = \frac{5}{4}(x - 2.5).$ Simplify: $y = \frac{5}{4}x - \frac{25}{8} + 2 = \frac{5}{4}x - \frac{9}{8}.$
   • Perpendicular bisector of $AC$ (passing through $(-1, 3.5)$): $y - 3.5 = -2(x + 1).$ Simplify: $y = -2x - 2 + 3.5 = -2x + 1.5.$

6. Solve the two equations to find the intersection point:

   • From $y = \frac{5}{4}x - \frac{9}{8}$ and $y = -2x + 1.5$, equate: $\frac{5}{4}x - \frac{9}{8} = -2x + 1.5.$
   • Clear fractions by multiplying through by 8: $10x - 9 = -16x + 12.$ Simplify: $26x = 21 \implies x = \frac{21}{26}.$
   • Substitute $x = \frac{21}{26}$ into $y = -2x + 1.5$: $y = -2\left(\frac{21}{26}\right) + 1.5 = -\frac{42}{26} + \frac{39}{26} = -\frac{3}{26}.$

Thus, the circumcenter of the triangle is: $\boxed{\left(\frac{21}{26}, -\frac{3}{26}\right)}.$

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Follow-up Questions:

1. What are the properties of the circumcenter in different types of triangles?
2. Can the circumcenter fall outside the triangle? If yes, when does that occur?
3. How is the circumradius related to the circumcenter?
4. How do you find the circumcenter of a triangle in 3D space?
5. How does the circumcenter relate to the Euler line of a triangle?

Tip: Always double-check calculations when solving equations with fractions to avoid arithmetic errors!