Euler line

• Euler's line

Named for Swiss mathematician Leonhard Euler.

Euler line (plural Euler lines)

1. (geometry) A line that, for a given triangle, passes through several important points, including the circumcentre, orthocentre and centroid; an analogous line for certain other 2- and 3-dimensional geometric figures.
   • 1983, The American Mathematical Monthly, volume 40, page 199:

     Since the line joining the circumcenter and orthocenter of a triangle is its Euler line, we see that this parabola is the envelope of the Euler lines of the triangles A_i.

   • 2000, Alfred S. Posamentier, Making Geometry Come Alive: Student Activities and Teacher Notes‎, page 147:

     The Euler line in the preceding figure is OH. N, the center of the nine-point circle, not only lies on the Euler line, but is also its midpoint.

   • 2011, Derek Allan Holton, A Second Step to Mathematical Olympiad Problems, World Scientific Publishing, Mathematical Olympiad Series, Volume 7, page 57,

     Show that the perpendicular bisector of LM in Figure 2.5 meets the Euler line halfway between the orthocentre and the circumcentre of ΔABC.

2. (graph theory) An Eulerian path, a looped path through a graph that passes along every edge exactly once.
   • 1961, Yale University, Graphs and Their Uses‎, page 25:

     Theorem 2.1 A connected graph with even local degrees has an Euler line.

   • 2009, J. P. Chauhan, Krishna's Applied Discrete Mathematics‎, page 279:

     In defining an Euler graph, some authors drop the requirement that the Euler line be closed.

   • 2013, Bhavanari Satyanarayana, Kuncham Syam Prasad, Near Rings, Fuzzy Ideals, and Graph Theory, page 372:

     Euler lines mainly deal with the nature of connectivity in graphs. The concept of an Euler line is used to solve several puzzles and games. […] A closed walk running through every edge of the graph G exactly once is called an Euler line.

• Hamiltonian path