In the image, I have constructed a nonagon (in Adobe Illustrator) by making a vertical equilateral triangle and rotating it by 40 degrees twice (i.e. three large equilateral triangles to form the nonagon). The point of rotation was set at a height of nine of the smallest triangles up from the bottom of the main 'upright' equilateral triangle, and centered (obviously).
As far as I can tell, this should result in a nonagon with equal length sides. As the image shows, it almost works, but, as circled, the smaller triangles do not quite align. I am 99.9% sure that this is not an error in constructing this image (two very talented friends have tried in separate attempts and come to the same results), but rather a mathematical error, or lack in my understanding.
What am I doing wrong, and how to fix this? Many thanks in advance.
I didn't count, but presumably you have $27$ small triangles on the side of each large equilateral triangle. The centroid of a triangle is $\frac 13$ of the way from a side to the opposite vertex, so counting $9$ small triangles up should give you the exact centroid. The misalignment you cite comes because the sides of the nonagon are not exactly $\frac 7{27}$ of the side of the equilateral triangle. You have not given any reason to expect that they should be. If you want to check, you can scale things so the top point of your nonagon is $(0,1)$ and the left point of the same triangle is $(\frac {\sqrt 3}2, -\frac 12)$. The upper right point is $(\cos 50^\circ,\sin 50^\circ)$ and the leftmost point is $(\cos 170^\circ, \sin 170^\circ)$ Use the two-point forms of the two lines and find the point of intersection, then find the distance from that point to $(1,0)$ I bet it is not exactly $\frac 7{27}$ of the equilateral triangle side.