



 
The simple answer is that there is no 'incircle' for many quadrilaterals, in the sense of a circle which meets all four sides, and there may not be a unique answer, if you shift your definition to meets three sides (think of a rectangle). This does make it difficult to 'find a formula'. I guess it depends what you want to do with it. Worst case, you could find incircles for triangles from sets of three lines, and then check which of them are suitable! Walter Whiteley
We are now into 'computational geometry' (a very interesting field).
Here is a site with a number of related computations
http://www.soe.ucsc.edu/~jcortes/software/packages/PlanGeom.m It turns out that finding circles with various properties around
collections of polygons or points is part of what people do (related First  is the polygon convex? This will simplify things (see below). As long as it is simple (no crossings) there will be some answer, but it gets increasingly complex!. Let me answer as if it is convex. Now,from a simpler point of view: the solution will contact three
edges (or at least one of them will if two of the sides are parallel
 e.g. the rectangle  and this can be shifted to contact three).
You could check all triples, find their incircles (or omit them if
the polygon interior is not 'inside' the triangle). The answer This is not 'fast' (there are nChoose3 such triples) so it would not
be good for, say, 1000 sides. I anticipate computational geometers
(who worry about worst case situations) could do better. Here is For example, you could start with a triple which contains the polygon, find an incircle, Now start testing the additional edges, one at a time, to see if this would cut off part of the incircle. If yes, then find the new triple (among just four) which is the largest incircle. Continue till all edges have been tested. This should take just n2 steps of checking for intersection, and perhaps selecting a new incircle (three more tests). Much more could be said, but I hope this helps. Walter Whiteley  


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