Don't Cross the Line

In order to minimize the chords in this case, we must maximize number of chords that cross the line.

Given $a$ and $b$ be the number of points on the left and right regions respectively, the number of chords that cross the line. equals $ab$ , where $a + b = 77$

And in order to maximize such product, it must be closest to a square form. Since $77$ is odd, the best we could do is taking $a,b$ as consecutive numbers. Thus, $ab = 39\times 38$ .

Then the total number of chords from all points = $\binom{77}{2} = 77 \times 38$ .

Finally, the least number of chords = $77\times 38 - 39\times 38 = 38^2 = \boxed{1444}$ .

Let there be $x$ points on the left half and $77-x$ points on the right half. Chords can only be formed using points on the same half of the circle, so the total number of chords formed is $c(x) = {x \choose 2} + {77-x \choose 2} = \frac{(x)(x-1)}{2} + \frac{(77-x)(77-x-1)}{2} = \frac{2x^2 - 154x + 5852}{2} = x^2 - 77x - 2926$

This will be minimized when its derivative is 0: $c'(x) = 2x - 77 = 0$ implies $x = 38.5$ . Since $x$ must be an integer, we check $38$ and $39$ : $c(38) = 1444\\c(39) = 1444$

Intuitively this makes sense: the number of chords is minimized when the points are split as close to half-and-half as possible. This means 38 on one side and 39 on the other (and it doesn't matter if it's 38 on the left or 38 on the right).

If there are more points on one side, we can reduce by shifting the points to other side. Therefore minimum points occur when those points are equal or it's plus 1.