Since the work of Malfatti, there has been a significant amount of work on methods for constructing Malfatti's three tangent circles; Richard K. Guy writes that the literature on the problem is "extensive, widely scattered, and not always aware of itself". Notably, presented a simple geometric construction based on bitangents; other authors have since claimed that Steiner's presentation lacked a proof, which was later supplied by , but Guy points to the proof scattered within two of Steiner's own papers from that time. Solutions based on algebraic formulations of the problem include those by , , , , and . The algebraic solutions do not distinguish between internal and external tangencies among the circles and the given triangle; if the problem is generalized to allow tangencies of either kind, then a given triangle will have 32 different solutions and conversely a triple of mutually tangent circles will be a solution for eight different triangles. credits the enumeration of these solutions to , but notes that this count of the number of solutions was already given in a remark by . The problem and its generalizations were the subject of many other 19th-century mathematical publications, and its history and mathematics have been the subject of ongoing study since then.