In any subspace of the real line Rwith the usual Euclidean metric d(x, y) = Ix-Yl, every triangle is degenerate. In R2 or R3 with the usual Euclidean metrics, a triangle is degenerate if and only if its are collinear. With our intuition of a degenerate triangle having collinear vertices extended to arbitrary metric spaces, we might expect that a metric space in which every triangle is degenerate must be linear. It might be reasonable to expect that any linear metric space is isometric to a subset of R with the usual metric. When classifying all metric spaces that have only degenerate triangles, we find that there are such metric spaces other than (isometric images of) subspaces of R. These other spaces, however, all have precisely four points and all are of the same form. In the final section, we illustrate that the usual topology on R2 can not be generated by any metric in which all triangles are degenerate. metric space (M, p) is a set of points M with a metric, or distance function, p: M x M o [O, oo) that satisfies some natural properties we expect of distances: p(x,y) = p(y,x) foranyx,y cM, p(x,y) = O if andonlyif x =y, and p(x, y) + Ply, z) 2 p(x, z) for any x, y, z c M (triangle inequality). If (M, p) and (N, b) are two metric spaces, a function f: M o N such that p(x,y) = b(f(x), f(y)) for any x,y c M is called an isometry. Isometries are always one-to-one. The metric spaces (M, p) and (N, b) are isometric if there exists an isometry from M onto N. Generally, one does not distinguish between isometric metric spaces. If (M, p) is a given metric space, we say that {x1, x2, X3} c M forms a degenerate triangle if p(xi, x;) + p(xj, Xk) = p(xi, Xk) for some permutation i, j, k of the indices {1, 2, 3}. Thus, {x1, x2, X3} forms a degenerate triangle if the triangle inequality is an equality for some permutation of the points x1, x2, and X3. We want to study metric spaces in which every triangle is degenerate, that is, metric spaces such that every 3-element subset forms a degenerate triangle. For brevity, such a metric space is called a degenerate space, and its metric is called a degenerate metric. Triangles with fewer than three distinct are clearly degenerate, and in what follows, we assume that all triangles have three distinct vertices, unless otherwise noted. With this understanding, we can make statements such as A four-point space has exactly four distinct triangles. The real line with the usual Euclidean metric d(x, y) = Ix-yl is denoted by R. It is a familiar fact that R is a degenerate space. Any subspace of a degenerate space is a degenerate space. Given any degenerate space M, one might suspect that it must be isomorphic to a subspace of R, and might attempt to construct an isometry from M into R. As we will see, the construction of such an isometry requires that if {y, o, p} and {o, p, x} are (degenerate) triangles in M with longest sides {y, p} and {o, x} respectively, then triangle {y, o, x} must have longest side {y, x}. Surprisingly, this