Given any $L = (\ell_1,\ell_2,\ldots,\ell_n)$ edge lengths, it is possible to
construct a cyclic (inscribed) convex polygon. This can be seen by viewing the
edges are rigid bars and the vertices as universal joints. Then place a polygonal
chain with those lengths in a large-radius circle, and shrink the radius until the
chain closes to a polygon:
However, the same proof idea does not work for tangential polygons.
A tangential polygon (circumscribed)
has each edge tangent to a circle:
(Image by Claudio Rocchini in Wikipedia.)
My question is:
Q. Which length sequences $L = (\ell_1,\ell_2,\ldots,\ell_n)$ can be realized
as the edge lengths of a tangential polygon of $n$ vertices?
In other words, characterize the realizable $L$.
For tangential quadrilaterals, $\ell_1+\ell_3 = \ell_2+\ell_4$, and
this generalizes for all even $n$.
But I am not sure that this necessary condition is also sufficient.
It is for $n=4$, but what is known for even $n$ greater than $4$?
Nor do I know of conditions for odd $n$.
Best Answer
This problem is discussed in Djukić, Janković, Matić, Petrović, The IMO Compendium, pg 561, extracted below.
If the edge lengths satisfy a certain necessary and sufficient condition (basically a sanity check), then (much as for the cyclic polygons described in the question) the corresponding polygon can be found by wrapping the sequence of edges tangentially along an arc of a large circle and then shrinking the circle until the first and last edges meet up.