Let A(XYZ) be the area of the triangle XYZ. A non-regular convex polygon P1P2…Pn is called guayaco if exists a point O in its interior such that A(P1OP2)=A(P2OP3)=⋯=A(PnOP1).Show that, for every integer n≥3, a guayaco polygon of n sides exists.
Problem
Source: Cono Sur Olympiad 2017, problem 2
Tags: geometry, cono sur
18.08.2017 03:05
Take a regular n-gon and apply a non-trivial affine transformation to it. Done!
18.08.2017 04:11
CantonMathGuy wrote: Take a regular n-gon and apply a non-trivial affine transformation to it. Done! And every affine transformation scales area, right?
18.08.2017 04:56
Affine transformations preserve ratios of areas.
29.08.2017 22:39
How did you solve this problem?
30.08.2017 21:17
If n=3, choose any non equilateral triangle and let O be the barycenter. If n=4, choose a paralelogram that isn't a square and let O be the intersection of the diagonals. For n≥5, take n−2 congruent isosceles triangles with an angle of \frac{180º}{n-2} in such a way that they add up to a flat angle. Then add 2 rectangle triangles with matching areas at the bottom to finish the polygon: