Vectors and the geometry of a triangle

Extract

MacNab [1] has recently considered the Euler line of a triangle, and derived expressions for the position vectors of certain important points like the orthocentre, the circumcentre and the incentre. In this note I show how these results, and some other ones, can be derived by a more direct use of vector methods than MacNab employed. I believe that such applications provide excellent material for use in a first course on vectors since only elementary properties are needed. Nevertheless, the power of vector methods is well illustrated and, in addition, the results are interesting and the symmetry of the algebra is very appealing. While I suspect that much of the material is not original–for example some of the results are in the well-known book by Weatherburn [2]–I am not aware of the existence of a single concise treatment, and I believe this has potential value.