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One-seventh area triangle

 

 
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One-seventh area triangle
 
 
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In plane geometry
Plane geometry

In mathematics, plane geometry may mean:*geometry of a plane ,*geometry of the Euclidean plane,or sometimes a plane is any flat surface that extends without end in all directions....
, a triangle ABC contains a triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 of one-seventh area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 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
Hugo Steinhaus

Wladyslaw Hugo Dionizy Steinhaus was a Poland mathematician and educator....
 is that the (central) triangle with sides p,q,r be reflected in its sides and vertices.






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In plane geometry
Plane geometry

In mathematics, plane geometry may mean:*geometry of a plane ,*geometry of the Euclidean plane,or sometimes a plane is any flat surface that extends without end in all directions....
, a triangle ABC contains a triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 of one-seventh area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 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
Hugo Steinhaus

Wladyslaw Hugo Dionizy Steinhaus was a Poland mathematician and educator....
 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

Martin Gardner is a popular American mathematics and science writer specializing in recreational mathematics, but with interests encompassing magic , pseudoscience, literature , philosophy, scientific skepticism, and religion....
 through James Randi
James Randi

James Randi is a Magician and Scientific skepticism best known as a challenger of paranormal claims and pseudoscience. Born Randall James Hamilton Zwinge,...
’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
Richard Feynman

Richard Phillips Feynman was an United States physicist known for the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as work in particle physics ....
 in a dinner conversation; they go on to give four different proofs.

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

In geometry Routh's theorem states the following: Let ABC be a triangle with area [ABC]. Let F, D and E be points in the sides AB, BC and AC such that the ratios AF/BF, BD/CD and CE/AE are r, s and t respectively....
.