Math Is Fun Forum

JohnnyReinB

Member

Registered: 2007-10-08

Posts: 453

Lines Passing through a Circle...

Can anyone prove this:

N = (x^2+x)/2 + 1

with x=number of lines passing through a circle

N= the maximum number of regions created with sides made of the lines and/or the edge of the circle

E.G.

with one line...

(1^2+1)/2 + 1 = 2 regions

with 2 lines:

(2^2+2)/2 + 1 = 4 regions

with 5 lines...

(5^2+5)/2 + 1 = 16 regions

I can see that this is true up to 6 lines by drawing them, but is it true up to, I dunno, 100 lines?

P.S.

How do you upload an image?

Last edited by JohnnyReinB (2008-02-15 23:32:29)

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"There is not a difference between an in-law and an outlaw, except maybe that an outlaw is wanted"

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Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Lines Passing through a Circle...

Should be possible with induction. It is true for x=1.

If you add one line, it can max pass through each line one time each. when the line intersects two lines in a row, it splits the area in between into two pieces. When you include the circle, the line will intersect x+2 times, ie dividing x+1 areas into two pieces each, which means an increase with x+1 areas.

also, consider a line AB which intersects all other lines in points p1,p2...pn. If you add one line CD and let it intersect line AB in a point different from p1,p2...pn and let its intersections with the circle be closer to ABs intersections with the circle than any other line, the line CD must pass through all lines (easy to vizualize or show with a figure). this shows that it is always possible to add a line that intersects all the other lines.

assume the formula holds for any x. then, replacing x with x+1, we get:

ie an increase of x+1 areas, and by the the induction principle, this holds for all x.

Last edited by Kurre (2008-02-16 02:58:37)