Prove for $n=1,2$ that if $$\frac{1}{(R+d)^n}+\frac{1}{(R-d)^n}=\frac{1}{r^n}$$ where $R,r$ are the radii of two circles with the smaller one being inside the larger one and $d$ being the distance between their centers, then there is a $n+2$ - polygon inscribed in the smaller one and inscribing the larger one.
Is this true for $n\geq 3$?
We know using Poncelet's porism it is enough to prove this happens for a polygon symmetric around the axis between the two centers and using that I've been able to calculate $n=1$ but $n=2$ was too complicated.
Is there a simpler way to see if it's correct?
The following figure shows a bicentric quadrilateral that is symmetric about the line of centers $CO$, is inscribed in circle $(C)$, and touches the circle$(O)$.
The following proof is excerpted from Johnson, Advanced Euclidean Geometry, 1929. $R,r$ have been renamed to $r,\rho$ and $d=\overline{CO}$ (the length of segment $CO$). The proof builds a system of similar right angled triangles and then ratio chases to derive the formula.
Note that in the second last line the expression $OD=\dfrac{\rho^2}{r+d}$ comes from $OD\cdot OA_1=\rho^2$, which can be easily demonstrated using similar triangles. $A_1$ is the inverse of $D$ in circle $(O)$. Similarly for the other expression.