Tutte's Triangle Lemma proves that if {a, b, c} is a triangle in a 3-connected matroid and neither M\a nor M\b is 3-connected, then M has a triad that contains a and exactly one of b and c. Hence {a, b, c} is contained in a fan of M with at least four elements. In this paper we ask for a somewhat stronger conclusion. When is it that, for each t in { a, b, c}, either M\t is not 3-connected, or M\t has a 3-separation that is not equivalent to a 3-separation induced by M? The main result describes the structure of M relative to { a, b, c} when this occurs. This theorem generalizes a result of Geelen and Whittle for sequentially 4-connected matroids. The motivation for proving this result was for use as an inductive tool for connectivity results aimed at representability questions. In particular, Geelen, Gerards, and Whittle use it in their proof of Kahn's Conjecture for 4-connected matroids.