Constructing a Poncelet 8-gon

Move the slider and the green points.

Here you can see how to construct a Poncelet 8-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 8-gon. The construction proceeds in two steps. First point 7 must be constructed then, from this point we construct a center of the consfoguration. From that we symmetrically construct the remaining two points 6 and 8. The construction mainly needs join and meet operations. However, some of the lines have to be intersected with the supporting conic. These intersections have to be made in a consistent way.



This is the construction:

  1. start with points $1,2,3,4,5$
  2. construct a conic $\mathcal{C}$ through these points
  3. $O=(1 \vee 2)\wedge (4\vee 5)$
  4. $P=(1 \vee 4)\wedge (2\vee 5)$
  5. $Q=(3 \vee 4)\wedge (3\vee Q)$
  6. $R=(1 \vee 5)\wedge (O\vee P)$
  7. $S=(1 \vee 5)\wedge (O\vee Q)$
  8. intersect $R \vee P$ with $\mathcal{C}$ to get $7$
  9. $M=(1 \vee 5)\wedge (3\vee 7)$
  10. intersect $M \vee 2$ with $\mathcal{C}$ to get $6$
  11. intersect $M \vee 4$ with $\mathcal{C}$ to get $8$