Some years ago, I posted two geometrical problems related with side-ratios in triangles and squares. This year, I was on my bed thinking when I remember this two problems and I try to generalize for regular polygons of n-sides. After some calculations in Geogebra, I found an interesting relation between the regular polygon (the original one) and the builded polygon which was a linear equation. The question from which the problem emerges is:
Let be a polygon of n sides (original) with vertex labeled from to . Between there is a point such , between there is a point such , between there is a point such and so on until complete the n vertex (builded).
Then, polygons' areas are plotted (axis- for the area of builded polygons and axis- for the area of the original polygons) to values from to
A linear equation is got it from the previous question.
For polygons of side one I got the areas in te table below which its graph
I did the same process for polygons of sides to and get the same linear equation. I'm intrigued about:
Why this data shows a linear equation?
How can I get a matemathical process to calculate the builded area?
Why the linear equations is so close from ?
Those are my main questions, I hope someone get interested in this problem and together we could work together on this.