# Necessary and sufficient condition for tangential polygon to be cyclic

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 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 if and only if $$H_1,H_2,\ldots ,H_n$$ are concyclic.

Picture for the case $$n=6$$:

GeoGebra applets that demonstrate this claim can be found here , here and here.

• what is concyclic? Mar 13, 2021 at 12:05
• @DimaPasechnik Concyclic points are the points that lie on the same circle. Mar 13, 2021 at 12:12

$$H_i$$ lies on the ray $$IA_i$$ and $$IH_i\cdot IA_i=2r^2$$ (where $$r=IB_i$$), since the midpoint of $$IH_i$$ is the midpoint of $$B_iB_{i-1}$$. Hence $$H_i$$ is the image of $$A_i$$ under the inversion with respect to center $$I$$ and radius $$\sqrt{2}r$$. Thus the result.