Constructing a Poncelet 7-gon

Move the slider and the green points.

Here you can see how to construct a Poncelet 7-gon from five free points. Move the white slider point down to unveil the construction step by step.

As long as no three points of the initial five points are collinear, the construction will lead to a Poncelet 7-gon. The construction mainly needs join and meet operations. However, at the very end two of the lines have to be intersected with the supporting conic. These intersections have to be made in a consistent way.

After the construction is completed the animation illustrates how to derive a Grünbaum-Rigby $(21_4)$-configuration from the seven points.

Experiment: Cycle one of the green points around another one and observe what happens.


This is the construction:

  1. start with points $1,2,3,4,5$
  2. construct a conic $\mathcal{C}$ through these points
  3. $O=(1 \vee 4)\wedge (2\vee 5)$
  4. $P=(1 \vee 3)\wedge (2\vee 4)$
  5. $Q=(3 \vee 5)\wedge (2\vee 4)$
  6. $R=(1 \vee 5)\wedge (O\vee P)$
  7. $S=(1 \vee 5)\wedge (O\vee Q)$
  8. intersect $R \vee Q$ with $\mathcal{C}$ to get $6$
  9. intersect $S \vee P$ with $\mathcal{C}$ to get $7$, (this instersection has to be chosen consistently to $6$).
  10. then $1,\ldots,7$ is a Poncelet 7-gon.