For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement amongst the following is:
There is a regular polygon with r/R = 2/3
There is a regular polygon with r/R = 1/√2
There is a regular polygon with r/R = 1/2
There is a regular polygon with r/R = √3/2

Let n be the number of sides of a regular polygon. Let a be the length of each side. Then, the radius of the inscribed circle (inradius) is given by r = a/(2tan(π/n)) and the radius of the circumscribed circle (circumradius) is given by R = a/(2sin(π/n)). Therefore, r/R = sin(π/n)/tan(π/n) = cos(π/n).

Let's check each option:

1. r/R = 2/3 => cos(π/n) = 2/3. This equation has a solution for n, as the range of the cosine function includes 2/3.
2. r/R = 1/√2 => cos(π/n) = 1/√2. This implies π/n = π/4, which means n = 4 (a square). This is a valid regular polygon.
3. r/R = 1/2 => cos(π/n) = 1/2. This implies π/n = π/3, which means n = 3 (an equilateral triangle). This is a valid regular polygon.
4. r/R = √3/2 => cos(π/n) = √3/2. This implies π/n = π/6, which means n = 6 (a regular hexagon). This is a valid regular polygon.

The false statement is that there is no regular polygon with r/R = 2/3. While the equation cos(π/n) = 2/3 has a solution, there is no integer solution for n, representing the number of sides of a polygon. Therefore, there is no such regular polygon. This is because the value of n is not an integer which means it is not possible to have a regular polygon with r/R = 2/3.