Construct a convex polygon such that each of its sides has the same length as one of its diagonals and each diagonal has the same length as one of its sides, or prove that such a polygon does not exist.
Problem
Source: Canada RepĂȘchage 2018/4
Tags: geometry
09.04.2018 16:10
It doesn't exist.
09.04.2018 16:11
Prove that it doesn't.
09.04.2018 16:30
A convex quadrilateral will always have a diagonal longer than all of its sides. Adding sides equal to the length of the diagonal will create longer diagonals
09.04.2018 16:43
ythomashu wrote: A convex quadrilateral will always have a diagonal longer than all of its sides. Adding sides equal to the length of the diagonal will create longer diagonals Prove this.
09.04.2018 16:45
wu2481632 wrote: ythomashu wrote: A convex quadrilateral will always have a diagonal longer than all of its sides. Adding sides equal to the length of the diagonal will create longer diagonals Prove this. Prove that you have to prove this.
09.04.2018 16:46
Math-Ninja wrote: wu2481632 wrote: ythomashu wrote: A convex quadrilateral will always have a diagonal longer than all of its sides. Adding sides equal to the length of the diagonal will create longer diagonals Prove this. Prove that you have to prove this. Prove that you have to prove that you have prove this.
09.04.2018 16:50
@above, prove that you have to prove that you have to prove that you have prove this.
09.04.2018 16:59
Math-Ninja wrote: wu2481632 wrote: ythomashu wrote: A convex quadrilateral will always have a diagonal longer than all of its sides. Adding sides equal to the length of the diagonal will create longer diagonals Prove this. Prove that you have to prove this. he can't because I was actually wrong. take a equilateral triangle stuck to an obtuse triangle with an angle of 120 or more degrees. anyway, we will move to this existing thread that is non-spammy
10.04.2018 03:40
@above that isn't convex... anyway this is the first time i see HSO being spammy cross posting my solution from canada forum InCtrl wrote:
Diagram is attached. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy81L2U1OWQ5OGRhN2Y0YmQ0NTU1MDgwNDRlZTkzNTc5NzcyM2I1MzMzLnBuZw==&rn=U2NyZWVuIFNob3QgMjAxOC0wMi0wMiBhdCAxMC40MC4yOCBQTS5wbmc=
10.04.2018 03:54
Let the polygon have n sides then the no. of diagonal are n(n-3)/2 equating it with n give n=0 or n=5