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

, a triangle ''ABC'' contains a

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 ...

having one-seventh of the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

of ''ABC'', which is formed as follows: the sides of this triangle lie on

cevian In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovann ...

s ''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 Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) 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 Jan Kazimierz Unive ...

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 scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...

through

James Randi James Randi (born Randall James Hamilton Zwinge; August 7, 1928 – October 20, 2020) was a Canadian-American stage magician, author and scientific skeptic who extensively challenged paranormal and pseudoscientific claims. Rodrigues 2010p. ...

’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 in a dinner conversation; they go on to give four different proofs.R.J. Cook & G.V. Wood (2004) "Feynman's Triangle", ''Mathematical Gazette'' 88:299–302 A more general result is known as Routh's theorem.

References

{{Reflist *H. S. M. Coxeter (1969) ''Introduction to Geometry'', page 211, John Wiley & Sons. Objects defined for a triangle Articles containing proofs Area Affine geometry