One-seventh area triangle

The area of the small triangle is one-seventh of the area of the big triangle.

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

A more general result is known as Routh's theorem.

References

• R.J. Cook & G.V. Wood (2004) Feynman's Triangle Mathematical Gazette 88:299–302.
• H. S. M. Coxeter (1969) Introduction to Geometry, page 211, John Wiley & Sons.
• Hugo Steinhaus (1960) Mathematical Snapshots
• James Randi (2001) Proof by Martin Gardner
• Michael de Villiers (2005) Feynman's Triangle: Some Feedback and More Mathematical Gazette 89:107.