I want to build a quadrilateral with sides, for example, $2:3:5:4$, knowing that it also has an inscribed circle (which all 4 sides are tangent to). How can you do it for a general case, using only a compass and a straightedge? (I know that the sums of opposite sides should be equal for an inscribed circle to exist.)
(If there are multiple quadrilaterals with different angles that can be constructed, any one would work for me.)
Update: I want the polygon to be convex (no angles over 180°). Playing around in Geogebra, I found that all 4 angle bisectors intersect at one point, for any convex polygon.