In plane geometry, a triangle ''ABC'' contains 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 ...

having one-seventh of the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of ''ABC'', which is formed as follows: the sides of this triangle lie on cevians ''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 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 (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...

through James Randi’s on-line magazine), the pair-wise congruences of overhanging and missing pieces of ''ABC'' is evident. As seen in the graphical solution, six plus the original equals the whole triangle ''ABC''. TriangleOneSeventhAreaGraphicalSoln

An early exhibit of this geometrical construction and area computation was given by Robert Potts in 1859 in his Euclidean geometry textbook. According to Cook and Wood (2004), this triangle puzzled

Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...

in a dinner conversation; they go on to give four different proofs.R.J. Cook & G.V. Wood (2004) "Feynman's Triangle", ''

Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

'' 88:299–302 A more general result is known as

Routh's theorem In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and ...

. Also see

Marion Walter Marion Walter (July 30, 1928 – May 9, 2021) was an internationally-known mathematics educator and professor of mathematics at the University of Oregon in Eugene, Oregon. There is a theorem named after her, called Marion Walter's Theorem or jus ...

’s theorem.

References

{{Reflist * H. S. M. Coxeter (1969) ''Introduction to Geometry'', page 211,

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. Objects defined for a triangle Articles containing proofs Area Affine geometry