Timeline for Principal diagonals of octagon meet in a single point
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 24, 2021 at 14:06 | comment | added | Pedja | @RobertBryant Thanks. | |
| May 24, 2021 at 13:35 | comment | added | Robert Bryant | By the way, note that the above proof works for any $2m$-gon $P$ that is circumscribed about an ellipse $E_1$ and inscribed in another ellipse $E_2$: The $m$ lines joining opposite pairs of vertices are of $P$ are concurrent. | |
| May 24, 2021 at 12:09 | comment | added | Robert Bryant | A short proof is as follows: Consider the two ellipses, since there is one octagon inscribed in one and circumscribed about the other, it follows from Poncelet's Theorem that there is a circle of such octagons. Moreover, when one chooses (projective) coordinates so that the inner ellipse $E_1$ is $x^2+y^2=1$ and the outer ellipse $E_2$ is $x^2/a^2+y^2/b^2=1$, by symmetry of the 'translation map', the opposite vertex of the octagon passing through $(x,y)\in E_2$ is $(-x,-y)$, so the line joining them passes through $(0,0)$. QED | |
| May 24, 2021 at 3:47 | history | asked | Pedja | CC BY-SA 4.0 |