Count the total number of line segments formed by joining every pair of points

With $n$ distinct points on a circle, joining every pair gives $\binom{n}{2} = \frac{n(n-1)}{2}$ line segments in total.

Count the blue line segments

Adjacent points on a circle form the sides of the polygon determined by the $n$ points.

Hence, the number of blue line segments is exactly $n.$

Count the red line segments

All remaining segments are red, so $\text{red} = \binom{n}{2} - n.$

Use the condition that red and blue segments are equal

Given: $\binom{n}{2} - n = n$

So, $\binom{n}{2} = 2n$

Substitute $\binom{n}{2} = \frac{n(n-1)}{2}$: $\frac{n(n-1)}{2} = 2n$

Multiply by $2$: $n(n-1) = 4n$

Since $n \ge 2$, divide by $n$: $n-1 = 4$ $n = 5$

Conclusion

The required integer value is $\boxed{5}.$