Note that the same vector ( a + b)/3 can be obtained in a couple of ways. Yet another elementary proof is due to Rosemary Ramsey it is based on the observation that, assuming the medians are not concurrent, their points of intersection form a triangle whose area would be necessarily zero, which would lead to a contradiction.Ĭonsider two vectors a and b emanating from the same point O. In other words, D = M c, the midpoint of AB.Īs Dani points, his proof does not rely on the fact that the common point of the three medians divides each in the ratio 2:1, but rather obtains that ratio as a side effect. In any parallelogram, the diagonals bisect each other. With two pairs of parallel sides, the quadrilateral AHBG is a parallelogram. In ΔACH, M b is the midpoint of AC, while G is the midpoint of CH. In ΔBCH, M a is the midpoint of BC and also BH||GM a. Draw through B a line parallel to AM a and let it intersect CG at H. Extend CG beyond G and further beyond its intersection with AB at D. Let G be the point of intersection of the medians AM a and BM b. Smith Jewish Day School, Rockville, MD, which I came across in Mathematics Teacher, v 96, n 6, Sept. The fourth proof is due to Dani Rubinstein, Charles E. Since BC/M bM c = 2, we also have BG/M bG = 2 and CG/M cG = 2. All their sides are in the same proportion. The triangles BCG and M bM cG are similar. The proof ends as before.Ī third elementary proof was suggested by Scott Brodie. Therefore, GM b and GM C are twice as short as respectively GB and GC. This implies that the quadrilateral SRBC is a parallelogram, so that its diagonals CR and BS are halved by their common point G. Then M bM c is the mid-line in both triangles ABC and GRS, whence SR = BC and the lines are parallel. Similarly, CM c extends beyond M c to the length of GM c.
Since for two medians the points thus defined coincide, the same, by symmetry, is true for the remaining median.Īnother elementary proof starts with extending BM b beyond M b to the length of GM b. Since diagonals of a parallelogram are halved by the point of intersection, we also have M bG = GS and M cG = GR which together with GS = SB and GR = RC show that, on both medians, G stands twice as far from one end (the vertex) than from the other end (the mid-point.) This condition uniquely defines a point on a median. Furthermore, the two lines are parallel (being both parallel to BC.) Thus, the quadrilateral M bM cSR is a parallelogram. Let also R be the mid-point of GC and S the mid-point of GB. Let G denote the point of intersection of BM b and CM c. the line joining the mid-points of two sides of a triangle, is parallel to the third side and equals its half (see, Euclid, Elements, VI.2 and VI.4.) Therefore, from ΔABC, M bM c = BC/2.
Three medians of a triangle meet at a point - centroid of the triangle.īelow are several proofs of this remarkable fact.Ī mid-line, i.e. Using the standard notations, in ΔABC, there are three medians: AM a, BM b, CM c. In this case, g is called a \((\rho ,C)\)- coarse inverse of f.In a triangle, a median is a line joining a vertex with the mid-point of the opposite side. There are constants k, h(0) such that for all \(a,b,c,a',b',c'\in X\) we have (Bowditch ) A coarse median space is a triple, where ( X, d) is a metric space and is a ternary operator on X satisfying the following: 1.1 Bowditch’s definition of coarse median space Definition 1.1 We will provide the missing combinatorial framework by defining coarse median algebras. This prompts the question to what extent there could be a combinatorial characterisation of coarse medians mirroring the notion of a median algebra. In contrast, for a coarse median space the metric is an essential part of the data, as evidenced by the fact that almost any ternary algebra can be made into a coarse median space by equipping it with a bounded metric. The interaction between the geometry and combinatorics of a CAT(0) cube complex is mediated by the fact that the edge metric can be computed entirely in terms of the median. Coarse median spaces as introduced by Bowditch provide a geometric coarsening of CAT(0) cube complexes which additionally includes \(\delta \)-hyperbolic spaces, mapping class groups and hierarchically hyperbolic groups. Its power stems from the beautiful interplay between the non-positively curved geometry of the space and the median algebra structure supported on the vertices as outlined by Roller. Gromov’s notion of a CAT(0) cube complex has played a significant role in major results in topology, geometry and group theory.
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