1 . X1 X2 X3 ....X9 are nine points on the circumference of circle . Line segments are drawn connecting each pair of points.

How many line segments are drawn?.

2. X1 X2 X3 ....X9 are nine points on the circumference of circle . Line segments are drawn connecting each pair of points.

What is the largest number of different points inside the circle at which at least two of these line segments intersect? (Remember that the points are not necessarily evenly spaced around the circle.)

Darkside
Sep 11, 2018

#1**+2 **

Here's the first one....if we consider the points to be A, B ,C, D...etc, we can choose any two of the points without regard to order.....this will result in C(9,2) ⇒ 36 segments being drawn

CPhill
Sep 11, 2018

#2**+1 **

By looking at images of symmetrically placed points and the lines connecting them I see that the number of intersections in the interior of the circle given n points is given by

\(ni = 2^{n-2}-3,~\forall n \geq 4 \\ \text{and for }n=9\text{ this results in 125 intersections.}\)

have fun verifying this.

Rom
Sep 11, 2018

#4**+1 **

I had mathematica draw the lines for 5, 6, and 7, points.

Counted intersections by eye for each. Worked out the pattern.

Rom
Sep 12, 2018

#5**0 **

I still don't undetstand how you got that formula, but it's wrong.

The number of intersections inside a polygonal can't be larger than the number of diagonals squared and the formula for that number is a polynomial P(n) while your formula is an exponential formula.

According to this the formula is

(n^{4}-6n^{3}+11n^{2}-6n)/24

When n is odd and the polygonal is a regular polygonal. When substituting 9 for n we get 126, so the maximal number of intersections of diagonals inside the circle (inside the polygonal) is at least 126.

Maybe i misunderstood something, please correct me if i'm wrong.

Guest Sep 12, 2018

edited by
Guest
Sep 12, 2018