One-seventh Area Triangle

In plane geometry, a triangle ''ABC'' contains a triangle having one-seventh of the area 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 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''.

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