To find the coordinates of vertices $A$, $B$, and $C$, we use the midpoint connection. If $M$ is the midpoint of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$, its coordinates are:

$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

Let $A$, $B$, and $C$ be $(x_A, y_A)$, $(x_B, y_B)$, and $(x_C, y_C)$. Given $D(5, 1)$, $E(6, 5)$, and $F(0, 3)$ are midpoints of $BC$, $AC$, and $AB$, we write equations for $x$-coordinates:

Equation 1:

$$\frac{x_B + x_C}{2} = 5$$

Equation 2:

$$\frac{x_A + x_C}{2} = 6$$

Equation 3:

$$\frac{x_A + x_B}{2} = 0$$

Step 1: Find the sum of $x$-coordinates.

Adding the three equations:

$$\frac{2x_A + 2x_B + 2x_C}{2} = 5 + 6 + 0$$

This simplifies to:

$$x_A + x_B + x_C = 11$$

Step 2: Calculate individual $x$-coordinates.

Subtracting each equation from the sum gives:

$$x_A = 11 - 2(5) = 1$$ $$x_B = 11 - 2(6) = -1$$ $$x_C = 11 - 2(0) = 11$$

Step 3: Find the sum and individual $y$-coordinates.

For $y$-coordinates:

$$\frac{y_B + y_C}{2} = 1$$ $$\frac{y_A + y_C}{2} = 5$$ $$\frac{y_A + y_B}{2} = 3$$

Adding gives:

$$y_A + y_B + y_C = 9$$

Subtracting each gives:

$$y_A = 9 - 2(1) = 7$$ $$y_B = 9 - 2(5) = -1$$ $$y_C = 9 - 2(3) = 3$$

∴ The vertices are $A(1, 7)$, $B(-1, -1)$, and $C(11, 3)$.