Introduction:
The question asks us to find the coordinates of the incentre of the triangle formed by the points (0,0), (0,3), and (4,0). The incentre of a triangle is the point where the angle bisectors of the triangle intersect. To find the coordinates of the incentre, we need to determine the equations of the angle bisectors and find their point of intersection.

Method:
To find the coordinates of the incentre, we will follow these steps:

Step 1: Find the lengths of the sides of the triangle:
We can find the lengths of the sides of the triangle using the distance formula. Let's call the points (0,0), (0,3), and (4,0) as A, B, and C, respectively.

The length of side AB = √[(x2 - x1)^2 + (y2 - y1)^2] = √[(0 - 0)^2 + (3 - 0)^2] = 3

The length of side BC = √[(x2 - x1)^2 + (y2 - y1)^2] = √[(4 - 0)^2 + (0 - 3)^2] = 5

The length of side AC = √[(x2 - x1)^2 + (y2 - y1)^2] = √[(4 - 0)^2 + (0 - 0)^2] = 4

Step 2: Find the equations of the angle bisectors:
The angle bisectors of a triangle divide the angles into two equal parts. To find the equations of the angle bisectors, we need to find the slopes of the sides of the triangle.

The slope of side AB = (y2 - y1) / (x2 - x1) = (3 - 0) / (0 - 0) = undefined (as the denominator is zero)

The slope of side BC = (y2 - y1) / (x2 - x1) = (0 - 3) / (4 - 0) = -3/4

The slope of side AC = (y2 - y1) / (x2 - x1) = (0 - 0) / (4 - 0) = 0

The angle bisector of side AB will be perpendicular to side AB, passing through its midpoint.

Step 3: Find the midpoint of side AB:
The midpoint of side AB = ((x1 + x2) / 2, (y1 + y2) / 2) = ((0 + 0) / 2, (0 + 3) / 2) = (0, 1.5)

Step 4: Find the equation of the angle bisector of side AB:
Since side AB is vertical, the equation of the angle bisector of side AB will be a horizontal line passing through the midpoint of AB.

Thus, the equation of the angle bisector of side AB is y = 1.5.

Step 5: Find the equation of the angle bisector of side BC:
The slope of the angle bisector of side BC is the negative reciprocal of the slope of side BC.