Find side length of son-polygon.

Question

Take a regular n sided polygon having side length 1, where n is odd. Draw all diagonals of this polygon. Around the center, you will find a smallest regular polygon similar to bigger one. Tell this smallest one 'son polygon'. Find the side length of this son-polygon in terms of n. for example in pentagon, side length is $(\sin18^\circ/\sin54^\circ)$

Answer 1

Let $n = 2m+1$, and let the radius of the polygon's circumcircle be $r$.

The sides of the smallest polygons are sub-segments of the longest diagonals. Connecting the center to the endpoints of one of these longest diagonals creates a central angle of size $\frac{2\pi m}{2m+1}$; dropping a perpendicular from the center to the diagonal bisects this angle, creating a right triangle with hypotenuse $r$ and "short leg" $r \cos\frac{\pi m}{2m+1}$. This is the "apothem" of the small triangle.

Connecting the center of the polygon to two adjacent vertices creates a central angle of size $\frac{2\pi}{2m+1}$. As above, we can compute the apothem of the large polygon to be $r\cos\frac{\pi}{2m+1}$.

Apothems of similar polygons are proportional to the side lengths: $$\frac{\text{side of small polygon}}{\text{side of large polygon}}=\frac{\text{apothem of small polygon}}{\text{apothem of large polygon}}$$ so, $$\text{side of small polygon} = \frac{\cos\frac{\pi m}{2m+1}}{\cos\frac{\pi}{2m+1}} =\frac{\sin\left(\frac{\pi}{2}-\frac{\pi m}{2m+1}\right)}{\sin\left(\frac{\pi}{2}-\frac{\pi}{2m+1}\right)} = \frac{\sin\frac{\pi}{2n}}{\sin\frac{\pi(n-2)}{2n}}$$

Answer 2

This is a layout for what to do.

1. Find your radius of parent polygon and take its circumcentre as origin.
2. You can easily label all points by taking any one arbitrarily at x-axis.
3. For easiness, I will name points as $r,r\alpha,r\alpha^2...r\alpha^{n-1}$ where $\alpha$ is our well loved $n$th root of unity.
4. Connect $r$ with $r\alpha^{\lfloor\frac{n}{2}\rfloor}$. Name this line 1.
5. Connect $r\alpha$ and $r\alpha^{\lfloor\frac{n}{2}\rfloor+1}$. Name this line 2.
6. I seem to have forgotten Connie's rotation method in argand plane. But, you can use the good vectors to find their intersection point. Write equation of line in vector form and find out their point of intersection.
7. Its distance from centre is known now. Now, use trigonometry again to find length of child polygon.