Let $ A_1A_2...A_n$ be a convex polygon. Show that there exists an index $ j$ such that the circum-circle of the triangle $ A_j A_{j + 1} A_{j + 2}$ covers the polygon (here indices are read modulo n).
Problem
Source: Indian postal coaching 2008
Tags: geometry, circumcircle, induction, extremal principle, combinatorics unsolved, combinatorics
22.12.2008 05:36
22.12.2008 12:37
I don't think that hint will work for me. I mean.... how will you show that that circle will always become some triangle's circumcircle?
23.12.2008 14:07
i would say try to find a triangle (general) whose circumcircle covers the n gon and then try to create more triangles
24.12.2008 11:53
aakansh92 wrote: Let $ A_1A_2...A_n$ be a convex polygon. Show that there exists an index $ j$ such that the circum-circle of the triangle $ A_j A_{j + 1} A_{j + 2}$ covers the polygon (here indices are read modulo n).
24.12.2008 13:59
Please be more clear and give full solution. Gaurav, could you do that problem during the postals? If yes, I am eager to look at the solution.
24.12.2008 14:02
Aaknash, were u at the IMOTC last year?
24.12.2008 17:55
Here's my (not well-written ) solution. Hope you understand the idea.
Do reply if you understand or not.
24.12.2008 21:30
Ah... Sorry... When I replied, I was thinking in another problem. Akashnil, I think you should replace three of your $ A_{j + 1}$'s by $ A_{j + 2}$'s. Anyway, what is wrt?
24.12.2008 21:50
wrt = with relation to, or something similar.
25.12.2008 07:28
wrt=with respect to. $ A$ is opposite $ B$ wrt $ CD$ means the points $ A$ and $ B$ lie on oposite sides of the line $ CD$. Yes I messed up some of $ A_{j+1}$ and $ A_{j+2}$s . Sorry. I've edited now. Thanks. And once again do reply if you understand it now or not.
25.12.2008 08:28
aakansh92 wrote: Please be more clear and give full solution.
. please verify the solution.
25.12.2008 08:44
Akashnil wrote: I don't know if the following requires proof. Lemma 1: Suppose $ S$ is a circle which contains $ Y,Z$ but doesn't contain $ X$. Then the portion of the circumference of $ S$ that lies opposite $ X$ wrt $ YZ$ lies outside the circumcircle of $ XYZ$ P.S.- above lemma can be easily proved since two circles(not identical) can intersect at only two points. P.P.S.- AKashnil, your proof is really very good.
25.12.2008 10:02
Yours too
25.12.2008 18:36
those two solutions are god damn BEAUTIFUL hats off to you both ith_power and Akashnil my solution amounted to atleast 5 whole written pages in Ideas but yours(both) is just beauty
27.12.2008 07:28
That's really wonderfull.Thanks!
31.12.2008 20:46
It is very unfortunate that this problem was taken from the very well known Arthur Engel book's Extremal principle chapter. Infact it is an example problem. Too bad of the problem setters in India. Anyway, akashnil's solution is superb as it is a different one . ith_power ur solution is superb as u have used the extremal principle well. I also solved it by that way and even in the book they solved it in the same way. Moderators if this is an unnecessary post sorry but just wanted to say that it was copied thats all.
01.01.2009 09:47
engel does not state consecative
01.01.2009 13:01
Enegel does state consecutive.