The triangle $ABC$ lies on the coordinate plane.  The midpoint of $\overline{AB}$ has coordinates $(-16, -63)$, the midpoint of $\overline{AC}$ has coordinates $(13, 50)$, and the midpoint of $\overline{BC}$ has coordinates $(6, -85)$.  What are the coordinates of point $A$?
1. Let the coordinates of point \( A \) be \( (x_1, y_1) \), the coordinates of point \( B \) be \( (x_2, y_2) \), and the coordinates of point \( C \) be \( (x_3, y_3) \).

2. Given the midpoints of the sides of the triangle:
   - Midpoint of \(\overline{AB}\) is \((-16, -63)\)
   - Midpoint of \(\overline{AC}\) is \((13, 50)\)
   - Midpoint of \(\overline{BC}\) is \((6, -85)\)

3. Using the midpoint formula, we can set up the following equations for the \(x\)-coordinates:
   \[
   \frac{x_1 + x_2}{2} = -16 \implies x_1 + x_2 = -32
   \]
   \[
   \frac{x_2 + x_3}{2} = 6 \implies x_2 + x_3 = 12
   \]
   \[
   \frac{x_1 + x_3}{2} = 13 \implies x_1 + x_3 = 26
   \]

4. We now have the system of equations:
   \[
   x_1 + x_2 = -32
   \]
   \[
   x_2 + x_3 = 12
   \]
   \[
   x_1 + x_3 = 26
   \]

5. Subtract the second equation from the third equation:
   \[
   (x_1 + x_3) - (x_2 + x_3) = 26 - 12 \implies x_1 - x_2 = 14
   \]

6. Add this result to the first equation:
   \[
   (x_1 - x_2) + (x_1 + x_2) = 14 - 32 \implies 2x_1 = -18 \implies x_1 = -9
   \]

7. Similarly, we write the equations for \(y_1, y_2\), and \(y_3\):
   \[
   \frac{y_1 + y_2}{2} = -63 \implies y_1 + y_2 = -126
   \]
   \[
   \frac{y_2 + y_3}{2} = -85 \implies y_2 + y_3 = -170
   \]
   \[
   \frac{y_1 + y_3}{2} = 50 \implies y_1 + y_3 = 100
   \]

8. We now have the system of equations:
   \[
   y_1 + y_2 = -126
   \]
   \[
   y_2 + y_3 = -170
   \]
   \[
   y_1 + y_3 = 100
   \]

9. Subtract the second equation from the third equation:
   \[
   (y_1 + y_3) - (y_2 + y_3) = 100 - (-170) \implies y_1 - y_2 = 270
   \]

10. Add this result to the first equation:
    \[
    (y_1 - y_2) + (y_1 + y_2) = 270 - 126 \implies 2y_1 = 144 \implies y_1 = 72
    \]

11. Therefore, the coordinates of point \( A \) are:
    \[
    \boxed{(-9, 72)}
    \]