Locus of Point C:
The locus of point C is given by the equation 2x + 3y = 5. This equation represents a straight line in the Cartesian coordinate system.

Finding the Coordinates of Point C:
To find the coordinates of point C, we can substitute the x and y coordinates of C into the equation of the locus and solve for C. Let's denote the coordinates of C as (x, y).

Substituting the x-coordinate of C, we have:
2x + 3y = 5
2(1) + 3y = 5
2 + 3y = 5
3y = 5 - 2
3y = 3
y = 1

Substituting the y-coordinate of C, we have:
2x + 3y = 5
2x + 3(2) = 5
2x + 6 = 5
2x = 5 - 6
2x = -1
x = -1/2

Therefore, the coordinates of point C are (-1/2, 1).

Finding the Coordinates of the Centroid:
The centroid of a triangle is the point of intersection of its medians. To find the coordinates of the centroid of triangle ABC, we need to find the average of the x-coordinates and the average of the y-coordinates of points A, B, and C.

Coordinates of A: (1, 2)
Coordinates of B: (3, 4)
Coordinates of C: (-1/2, 1)

Average x-coordinate:
(1 + 3 + -1/2)/3 = (6 + 18 - 1)/6 = 23/6

Average y-coordinate:
(2 + 4 + 1)/3 = 7/3

Therefore, the coordinates of the centroid of triangle ABC are (23/6, 7/3).

Locus of the Centroid:
Since the coordinates of the centroid are (23/6, 7/3), the locus of the centroid can be represented by the equation obtained by substituting these coordinates into the equation of the locus of point C.

2x + 3y = 5
2(23/6) + 3(7/3) = 5
23/3 + 7 = 5
23/3 + 21/3 = 5
44/3 = 5

Therefore, the locus of the centroid of triangle ABC is given by the equation 2x + 3y = 5, which is the same as the locus of point C.

Firstly consider a point of centroid is (h,k), and assume point c(s,t)we know that cordinates of centroid are h=(3+1+s)/3 k=(4+2+t)/3by this we get the cordinates of point c (3h-4 ,3k-6 ) by satisfying in the locus of c == 2(3h-4)+3(3k-6)=5so we will get 6h+9k = 31so the locus of centroid of ABC is 6x+9y =31 that is the answer