Euler and triangle geometry

  title={Euler and triangle geometry},
  author={Gerry Leversha and G. C. Smith},
  journal={The Mathematical Gazette},
  pages={436 - 452}
There is a very easy way to produce the Euler line, using transformational arguments. Given a triangle ABC, let AʹBʹ'C be the medial triangle, whose vertices are the midpoints of the sides. These two triangles are homothetic: they are similar and corresponding sides are parallel, and the centroid, G, is their centre of similitude. Alternatively, we say that AʹBʹC can be mapped to ABC by means of an enlargement, centre G, with scale factor –2. 
8 Citations
95.18 Normals to the Euler line
  • J. Scott
  • Mathematics
    The Mathematical Gazette
  • 2011
95.18 Normals to the Euler line In both Cartesian and areal coordinates, the parallelism of lines is simply expressed by means of a null determinant but perpendicularity is rather more problematical
Miscellaneous triangle properties
  • J. Scott
  • Mathematics
    The Mathematical Gazette
  • 2010
In this note we will discuss five topics from triangle geometry and occasionally encounter something new. Areal coordinates (presented as Appendix A in [1]) will be used throughout, with the
Degree of Triangle Centers and a Generalization of the Euler Line
We introduce a concept "degree of triangle centers", and give a formula expressing the degree of triangle centers on generalized Euler lines. This generalizes the well known 2 : 1 point configuration
Circles in areals
August Möbius introduced the system of barycentric or areal coordinates in 1827 [1, 2]. The idea is that one may attach weights to points, and that a system of weights determines a centre of mass.
96.25 Going halfway with circular boundaries
Problem Corner
  • N.J.L.
  • Education
    The Mathematical Gazette
  • 2010
Problem Corner Solutions are invited to the following problems. They should be addressed to Nick Lord at Tonbridge School, Tonbridge, Kent TN9 lJP (e-mail: and should arrive
96.26 The Exeter point revisited
  • J. Scott
  • Mathematics
    The Mathematical Gazette
  • 2012


Euler's Triangle Determination Problem
We give a simple proof of Euler's remarkable theorem that for a non- degenerate triangle, the set of points eligible to be the incenter is precisely the orthocentroidal disc, punctured at the
Statics and the moduli space of triangles
The variance of a weighted collection of points is used to prove classi- cal theorems of geometry concerning homogeneous quadratic functions of length (Apollonius, Feuerbach, Ptolemy, Stewart) and to
The Algebra of Geometry
Projective geometry has always been called the geometry of joining and intersecting to contrast it with Euclidean, affine and other geometries which besides consider congruency, perpendicularity and
Advanced Euclidean Geometry
This book discusses Euclidean Geometry, an Elementary Treatise on the Geometry of the Triangle and the Circle and its Subgeometries, and its Applications.
Tritangent centers and their triangles
  • American Mathematical Monthly,
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Plane Euclidean geometry
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The algebra of geometry, Highperception
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A . P . Guinand , Tritangent centers and their triangles
  • American Mathematical Monthly
  • 1960