In plane geometry

Plane geometry

In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...

, a triangle ABC contains a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 of one-seventh area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

 of ABC formed as follows: the sides of this triangle lie on lines p, q, r where

p connects A to a point on BC that is one-third the distance from B to C,

q connects B to a point on CA that is one-third the distance from C to A,

r connects C to a point on AB that is one-third the distance from A to B.

The proof of the existence of the one-seventh area triangle follows from the construction of six parallel lines:

two parallel to p, one through C, the other through q.r

two parallel to q, one through A, the other through r.p

two parallel to r, one through B, the other through p.q.

The suggestion of Hugo Steinhaus

Hugo Steinhaus

Władysław Hugo Dionizy Steinhaus was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics...

 is that the (central) triangle with sides p,q,r be reflected in its sides and vertices. These six extra triangles partially cover ABC, and leave six overhanging extra triangles lying outside ABC. Focusing on the parallelism of the full construction (offered by Martin Gardner

Martin Gardner

Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...

 through James Randi

James Randi

James Randi is a Canadian-American stage magician and scientific skeptic best known as a challenger of paranormal claims and pseudoscience. Randi is the founder of the James Randi Educational Foundation...

’s on-line magazine), the pair-wise congruencies of overhanging and missing pieces of ABC is evident. Thus six plus the original equals the whole triangle ABC.

According to Cook and Wood (2004), this triangle puzzled Richard Feynman

Richard Feynman

Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

 in a dinner conversation; they go on to give four different proofs. De Villiers (2005) provides a generalization and an analogous result for a parallelogram

Parallelogram

In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

.

A more general result is known as Routh's theorem

Routh's theorem

In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the intersection of three cevians...

.