plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, the Morley centers are two special points associated with a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

. Both of them are

triangle center In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, ...

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 se ...

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. This resource is hosted at the University of Evansville The University of Evansville (UE) is a priv ...

, 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 celeb ...

in around 1899. FirstMorleyCenter

Definitions

Let be the triangle formed by the intersections of the adjacent angle trisectors of triangle . is called the ''Morley triangle'' of .

states that the Morley triangle of any triangle is always an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

First Morley center

Let be the Morley triangle of . 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 figure. The same definition extends to any object in n-d ...

of is called the ''first Morley center'' of .

Second Morley center

Let be the Morley triangle of . Then, the lines are concurrent. The point of concurrence is called the ''second Morley center'' of triangle .

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 ...

of the first Morley center of triangle are

\cos \tfrac + 2 \cos \tfrac \cos \tfrac : \cos \tfrac + 2 \cos \tfrac \cos \tfrac : \cos \tfrac + 2 \cos \tfrac \cos \tfrac

Second Morley center

The trilinear coordinates of the second Morley center are

\sec \tfrac : \sec \tfrac : \sec \tfrac

References

{{reflist Triangle centers