Towards a splitter theorem for internally 4-connected binary matroids III

In our quest to find a splitter theorem for internally 4-connected binary matroids, we proved in the preceding paper in this series that, except when M or its dual is a cubic Mobius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N either has a proper internally 4-connected minor M^' with an N-minor such that |E(M)-E(M^')|=<3 or has, up to duality, a triangle T and an element e of T such that M\e has an N-minor and has the property that one side of every 3-separation is a fan with at most four elements. This paper proves that, when we cannot find such a proper internally 4-connected minor M^' of M, we can incorporate the triangle T into one of two substructures of M: a bowtie or an augmented 4-wheel. In the first of these, M has a triangle T^' disjoint from T and a 4-cocircuit D^@? that contains e and meets T^'. In the second, T is one of the triangles in a 4-wheel restriction of M with helpful additional structure.