# One-seventh area 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 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.[1] 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*.[2]

An early exhibit of this geometrical construction and area computation was given by Robert Potts in 1859 in his Euclidean geometry textbook.[3]

According to Cook and Wood (2004), this triangle puzzled Richard Feynman in a dinner conversation; they go on to give four different proofs[4] De Villiers (2005) provides a generalization and an analogous result for a parallelogram.[5]

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

## References

- Hugo Steinhaus (1960)
*Mathematical Snapshots* - James Randi (2001) That Dratted Triangle, proof by Martin Gardner
- Robert Potts (1859)
*Euclid's Elements of Geometry*, Fifth school edition, problems 59 and 100, pages 78 & 80 via Internet Archive - R.J. Cook & G.V. Wood (2004) "Feynman's Triangle",
*Mathematical Gazette*88:299–302 - Michael de Villiers (2005) "Feynman's Triangle: Some Feedback and More"
*Mathematical Gazette*89:107

- H. S. M. Coxeter (1969)
*Introduction to Geometry*, page 211, John Wiley & Sons.

## External links

- Feynman's Triangle at Dynamic Geometry Sketches, an interactive dynamic geometry sketch with some generalizations as well.