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Exbicentric & Bicentric Quadrilateral

Ex-bicentric quadrilaterals are cyclic, while the extensions of their sides are tangent to another circle. I found almost no diagrams of such quadrilaterals published online, and in the rare instances where they do appear, a description of their construction is missing; I therefore conclude that no one knows how to construct them. However, I discovered that their construction is extremely simple: the extensions of the sides of an ex-bicentric quadrilateral are tangent to a circle at the points where it is intersected by two of its chords, the extensions of which intersect perpendicularly at a point outside the circle.

I created a GeoGebra application that allows for the transition from bicentric to ex-bicentric quadrilaterals by varying three parameters (accompanied by a description):

https://www.geogebra.org/m/nzcgugf9

Ex-bicentric quadrilaterals are governed by the same relationships that define bicentric quadrilaterals, and Fuss's relation applies to them as well. In this album, you will find a 3500×3500 resolution image illustrating interesting relationships between the parameters of a bicentric quadrilateral—relationships that also hold true for ex-bicentric quadrilaterals.

Added to Mathematical Research — 1 month ago