The given complex numbers are 1 + i, -2 + 3i, and 5/3 on the Argand plane. We need to determine whether these points represent the vertices of an equilateral triangle, an isosceles triangle, collinear points, or none of these.

To analyze this, let's plot these complex numbers on the Argand plane. The real part of a complex number represents its x-coordinate, and the imaginary part represents its y-coordinate.

Plotting the Complex Numbers:
- Complex number 1 + i is located at (1, 1) on the Argand plane.
- Complex number -2 + 3i is located at (-2, 3) on the Argand plane.
- Complex number 5/3 is located at (5/3, 0) on the Argand plane.

Now, let's analyze each case:

a) Vertices of an Equilateral Triangle:
For the points to form an equilateral triangle, they should be equidistant from each other. Let's calculate the distances between the points:

- Distance between (1, 1) and (-2, 3):
√[(1 - (-2))^2 + (1 - 3)^2] = √[9 + 4] = √13

- Distance between (-2, 3) and (5/3, 0):
√[(-2 - 5/3)^2 + (3 - 0)^2] = √[49/9 + 9] = √(49 + 81) = √130

- Distance between (5/3, 0) and (1, 1):
√[(5/3 - 1)^2 + (0 - 1)^2] = √[25/9 + 1] = √(25 + 9) = √34

The distances between the points are not equal, so they do not form an equilateral triangle.

b) Vertices of an Isosceles Triangle:
For the points to form an isosceles triangle, any two sides of the triangle should be equal in length. Let's calculate the lengths of the sides:

- Length of the side between (1, 1) and (-2, 3):
√[(1 - (-2))^2 + (1 - 3)^2] = √[9 + 4] = √13

- Length of the side between (-2, 3) and (5/3, 0):
√[(-2 - 5/3)^2 + (3 - 0)^2] = √[49/9 + 9] = √(49 + 81) = √130

- Length of the side between (5/3, 0) and (1, 1):
√[(5/3 - 1)^2 + (0 - 1)^2] = √[25/9 + 1] = √(25 + 9) = √34

The lengths of the sides are not equal, so they do not form an isosceles triangle.

c) Collinear Points:
To check if the points are collinear, we can calculate the slope of the line passing through any two points. If