mg.metric geometry – Collinearity in tangential pentagon

I am looking for a proof of the following claim:

Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of diagonals drawn from endpoints of that same side are collinear.

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The GeoGebra applet that demonstrates this claim can be found here.

mg.metric geometry – Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim?

Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from that vertex.Then, center of excircle of this triangle which touches side of pentagon, the vertex of pentagon opposite to that side and the incenter of the pentagon formed by diagonals are collinear.

enter image description here

GeoGebra applet that demonstrates this claim can be found here.

mg.metric geometry – Collinearity in bicentric polygons

Can you provide a proofs for the following two claims?

Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear.

Claim 2. The circumcenter, the incenter, and the incenter of the polygon formed by long diagonals in a bicentric odd-sided polygon are collinear.

Picture for the case when the polygon is pentagon:

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Picture for the case when the polygon is hexagon:

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GeoGebra applets for: quadrilateral, pentagon,hexagon,heptagon and octagon.

The case where the polygon is a quadrilateral is already known in the literature and the case where the polygon is a pentagon has been discussed here.