geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

the Morley centers are two special points associated with a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

. Both of them are

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...

s. One of them called first Morley center (or simply, the Morley center ) is designated as X(356) in

Clark Kimberling Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the l ...

, while the other point called second Morley center (or the 1st Morley–Taylor–Marr Center) is designated as X(357). The two points are also related to

Morley's trisector theorem In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem ...

which was discovered by

Frank Morley Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry. Among his mathematical accomplishments was the discovery and proof of the celebr ...

in around 1899. FirstMorleyCenter

Definitions

Let ''DEF'' be the triangle formed by the intersections of the adjacent angle trisectors of triangle ''ABC''. Triangle ''DEF'' is called the ''Morley triangle'' of triangle ''ABC''. Morley's trisector theorem states that the Morley triangle of any triangle is always an equilateral triangle.

First Morley center

Let ''DEF'' be the Morley triangle of triangle ''ABC''. The

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...

of triangle ''DEF'' is called the ''first Morley center'' of triangle ''ABC''.

Second Morley center

Let ''DEF'' be the Morley triangle of triangle ''ABC''. Then, the lines ''AD'', ''BE'' and ''CF'' are concurrent. The point of concurrence is called the ''second Morley center'' of triangle ''ABC''.

Trilinear coordinates

First Morley center

The

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

of the first Morley center of triangle ''ABC'' are : cos ( ''A''/3 ) + 2 cos ( ''B''/3 ) cos ( ''C''/3 ) : cos ( ''B''/3 ) + 2 cos ( ''C''/3 ) cos ( ''A''/3 ) : cos ( ''C''/3 ) + 2 cos ( ''A''/3 ) cos ( ''B''/3 ).

Second Morley center

The trilinear coordinates of the second Morley center are : sec ( ''A''/3 ) : sec ( ''B''/3 ) : sec ( ''C''/3 ).

References

{{reflist Triangle centers