In a coordinate plane, consider the points A(1, 2), B(5, 6), and C(x, y) such that triangle ABC is an equilateral triangle. If the coordinate C needs to be determined under the constraint that the side lengths of the triangle must be equal to the distance from point A to point B, what are the coordinates of point C?
The distance between points A and B can be calculated using the distance formula:
$$ d = ext{sqrt}((x_2 - x_1)^2 + (y_2 - y_1)^2) $$
Using the coordinates of A and B, calculate the side length and then deduce the location of C that keeps the triangle equilateral.