Chord lines

aka secant lines. Keeping things simple - let's say that all chord lines are different so no two chord lines here have more than one point in common and if they do that point is not on the circumference of the circle, and there are no intersection points where three or more chord lines meet, so intersections when they do occur are places where exactly two chord lines cross.

Given that, it is clear enough that

* 1 chord line divides the area of a circle into exactly two parts.
* 2 chord lines can have no intersection point; circle is cut in three parts.
* 2 chord lines can have an intersection point; circle is cut in four parts.
* 3 chord lines can have 0-3 intersection points, and divide the circle into 4-7 parts.

Is it true that there that there are at most c (c - 1) / 2 intersection points for c chord lines, and if there are p intersection points and and c chord lines, the area of the circle is cut into exactly 1 + p + c parts?

Solution. As a chord lines are added, it can cut all the chord lines before it; this can be done by choosing 2n points on the circumference, and points 1, ... n are the starting points for the 1st, 2nd .. n'th chort line, and points n+1, ... 2n are the ending points, in the same order. The first line has nothing to intercept, the second intercept the 1st, the 3rd the two preceding ones etc for a total of

p = 0+ 1 + 2 + 3 + ...+ c-1 = c (c - 1) / 2.

As for areas, there is one at first. Every line cuts through at least one area, dividing it in two, increasing the # of areas by one, so c lines add the # of areas by c. If there is an intersection point, the line passes through an additional area, adding the # of areas by one; p intersection points mean p additional areas, for a grand total of 1 + c + p.