Let's start by labeling the coordinates of the vertices of the given triangle.

To find the circumcenter, we need equations for the perpendicular bisectors of at least two sides of the triangle. Recall that a bisector cuts something in half, so we want perpendicular lines that perfectly divide the side of the triangle in half.

Finding Perpendicular Bisectors

By the Slopes of Perpendicular Lines Theorem, we know that horizontal and vertical lines are perpendicular. Since $\overline{D F}$ is horizontal, any perpendicular line will be vertical. Examining the diagram, we can also identify the midpoints of this side.

Given the information, we know that the perpendicular bisector through $\overline{D F}$ has the equation $x = 4 .$ To find the other perpendicular bisector, we need to find the equation of the line perpendicular to $\overline{D E},$ passing through the midpoint between the $D$ and $E .$ This will be a four-step process.

Find the slope of the line $D E,$
Find the slope of the line perpendicular to $D E,$
Find the midpoint between $D$ and $E,$
Find the equation of the perpendicular bisector.

Let's start with the first step.

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