Скачать или смотреть Bicentric Family IV: Constant-Perimeter Limiting Point Pedals in the N=4 Case

The video describes properties of the Poncelet Bicentric family of quadrilaterals interscribed by two circles which can be calculated via Fuss's formula for N=4 [1, Eqn. (39)]. Let l1,l2 denote the two limiting points [2] of the circumcircle-incircle pair of the family. A well know result about bicentric polygons is that their diagonals meet at one of the limiting points.

The video showcases a few curious properties of the bicentrics' pedal polygons with respect to either l1 and l2, to be sure:

a) the l1-pedal has constant perimeter and its vertices sweep a loopless Pascal limaçon.
b) all 4 vertices of the l2-pedal are dynamically collinear (zero area) and sweep a Pascal limaçon whose loop has a node at l2. The perimeter of the l2-pedal is also constant and its sum of cosines is invariant and equal to 4.

[1] E. Weisstein, "Poncelet Porism", MathWorld, 2021. https://mathworld.wolfram.com/Poncele...

[2] E. Weisstein, "Limiting Points, MathWorld, 2021. https://mathworld.wolfram.com/Limitin...