# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]Let $A(XYZ)$ be the area of the triangle $XYZ$. A non-regular convex polygon $P_1 P_2 \ldots P_n$ is called guayaco if exists a point $O$ in its interior such that$$A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).$$Show that, for every integer $n \ge 3$, a guayaco polygon of $n$ sides exists.

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" _i="1" _address="0.1.0.2.1" hover_enabled="0"]A regular polygon is obviously guayaco. As regular polygons are not allowed, we should try to "perturb" the polygon a little bit, preserving the areas. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" _i="2" _address="0.1.0.2.2" hover_enabled="0"]Let us look at transformations of the plane of the form $(x,y)\mapsto (ax+by,cx+dy)$. Show that, if $|ad-bc|=1$ then this transformation preserves areas. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" _i="3" _address="0.1.0.2.3" hover_enabled="0"]The class of transformations defined in hint 2 are known as area-preserving transformations. Clearly, taking the image of a regular $n$-gon under an area-preserving transformation should do the trick. However, there is a caveat. The resulting polygon might become concave.  [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27" _i="4" _address="0.1.0.2.4" hover_enabled="0"]To ensure that the resulting polygon remains convex, it suffices to take an area-preserving map close to the identity. Use $a=f(n),b=c=0,d=1/f(n)$ where $f(n)$ is a number dependent on $n$. The larger the value of $n$, the smaller $|1-f(n)|$ should be. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.26.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]