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To find the coordinates of the third vertex of the triangle given the centroid and two vertices, we can use the property of a centroid that states the centroid divides each median in a 2:1 ratio.

Let's denote the coordinates of the third vertex as (x, y).

The coordinates of the centroid are given as (5/3, 3).

Let's calculate the coordinates of the midpoint of the segment joining the third vertex and the first vertex [(x + 0)/2, (y + 1)/2]:

[(x + 0)/2, (y + 1)/2] = [(x/2), (y + 1)/2]

Now, according to the centroid property, the x-coordinate of the centroid is the average of the x-coordinates of the three vertices:

(x/2 + 0 + 2)/3 = 5/3

Simplifying this equation, we get:

(x/2 + 2)/3 = 5/3

Cross-multiplying, we have:

3(x/2 + 2) = 5

3x/2 + 6 = 5

3x/2 = 5 - 6

3x/2 = -1

3x = -2

x = -2/3

Now, let's calculate the y-coordinate of the centroid using the same approach:

[(x + 0)/2, (y + 1)/2] = [(x/2), (y + 1)/2]

According to the centroid property, the y-coordinate of the centroid is the average of the y-coordinates of the three vertices:

[(x/2) + 1 + 3]/3 = 3

Simplifying this equation, we get:

[(x/2) + 4]/3 = 3

Cross-multiplying, we have:

(x/2) + 4 = 9

x/2 = 9 - 4

x/2 = 5

x = 10

Therefore, the coordinates of the third vertex are (10, -2/3).