Can you prove or disprove the following claim?  

>**Claim**. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided [tangential polygon][1] and let $B_1,B_2, \ldots ,B_n$ be the contact points of the inscribed circle and polygon sides such that $B_1$ lies on $A_1A_2$, $B_2$ lies on $A_2A_3$ ,etc. Denote by $H_1,H_2, \ldots,H_n$  the orthocenters of the triangles $\triangle A_1B_1B_n$, $\triangle A_2B_2B_1$,....,$\triangle A_nB_{n}B_{n-1}$ . Then the polygon is [cyclic][2] if and only if $H_1,H_2,\ldots ,H_n$ are [concyclic][3].

Picture for the case $n=6$:

[![enter image description here][4]][4]

GeoGebra applets that demonstrate this claim can be found [here][5] , [here][6] and [here][7].


  [1]: https://en.wikipedia.org/wiki/Tangential_polygon
  [2]: https://mathworld.wolfram.com/CyclicPolygon.html
  [3]: https://en.wikipedia.org/wiki/Concyclic_points
  [4]: https://i.sstatic.net/SKmWh.png
  [5]: https://www.geogebra.org/m/dawjjn7x
  [6]: https://www.geogebra.org/m/wzapgznq
  [7]: https://www.geogebra.org/m/kgb4yza4