If the incentre of an equilateral triangle is $$(1, 1)$$ and the equation of its one side is $$3x + 4y + 3 = 0$$, then the equation of the circumcircle of this triangle is:

We are given that the triangle is equilateral, its incentre is at (1, 1), and the equation of one side is $$3x + 4y + 3 = 0$$. In an equilateral triangle, the incentre, circumcentre, centroid, and orthocentre all coincide at the same point. Therefore, the circumcentre is also at (1, 1).

The circumcircle has its centre at the circumcentre. So, the centre of the circumcircle is (1, 1). To find the equation of the circumcircle, we need its radius, which is the circumradius, denoted as $$R$$.

Since the incentre and circumcentre coincide, we can use the relationship between the inradius $$r$$ and circumradius $$R$$ for an equilateral triangle. The relationship is $$r = \frac{R}{2}$$. Therefore, $$R = 2r$$.

To find the inradius $$r$$, we compute the perpendicular distance from the incentre (1, 1) to the given side $$3x + 4y + 3 = 0$$. The formula for the distance from a point $$(x_0, y_0)$$ to a line $$ax + by + c = 0$$ is $$\frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$$.

Here, $$a = 3$$, $$b = 4$$, $$c = 3$$, $$x_0 = 1$$, $$y_0 = 1$$. Substituting these values:

$$ r = \frac{|3 \cdot 1 + 4 \cdot 1 + 3|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 4 + 3|}{\sqrt{9 + 16}} = \frac{|10|}{\sqrt{25}} = \frac{10}{5} = 2 $$

So, the inradius $$r = 2$$. Then, the circumradius $$R = 2r = 2 \cdot 2 = 4$$.

The equation of a circle with centre $$(h, k)$$ and radius $$R$$ is $$(x - h)^2 + (y - k)^2 = R^2$$. Substituting $$h = 1$$, $$k = 1$$, $$R = 4$$:

$$ (x - 1)^2 + (y - 1)^2 = 4^2 $$

Expanding the equation:

$$ (x - 1)^2 = x^2 - 2x + 1 $$

$$ (y - 1)^2 = y^2 - 2y + 1 $$

Adding these:

$$ x^2 - 2x + 1 + y^2 - 2y + 1 = 16 $$

Combining like terms:

$$ x^2 + y^2 - 2x - 2y + 2 = 16 $$

Subtracting 16 from both sides to bring the equation to standard form:

$$ x^2 + y^2 - 2x - 2y + 2 - 16 = 0 $$

$$ x^2 + y^2 - 2x - 2y - 14 = 0 $$

Comparing this with the given options:

A. $$x^2 + y^2 - 2x - 2y - 2 = 0$$

B. $$x^2 + y^2 - 2x - 2y + 2 = 0$$

C. $$x^2 + y^2 - 2x - 2y - 7 = 0$$

D. $$x^2 + y^2 - 2x - 2y - 14 = 0$$

The equation $$x^2 + y^2 - 2x - 2y - 14 = 0$$ matches option D.

Hence, the correct answer is Option D.