19 Jun '12 15:28

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?

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?