11403

Triangles with Equal Area Are Equidecomposable (Equivalent by Dissection)

A dissection of a polygon is a set of polygons that cover exactly without gaps or overlaps. Two polygons are equidecomposable (or equivalent by dissection) if there is a dissection of one that can be reassembled to form the other. In 2D, two polygons of equal area are equidecomposable. On the other hand in 3D, even the cube and the tetrahedron of equal volume are not equidecomposable.
The proof of the 2D case depends on the equidecomposability of two triangles of equal area, which can be split into two parts. The first part is a proof that two triangles with equal bases and equal altitudes are equivalent by dissection. This was proved in the Demonstration Two Equidecomposable Triangles.
It remains to prove that any two triangles of equal area are equidecomposable. So suppose that the two triangles and have the same area but unequal bases, and that . The aim is to construct a triangle , again of the same area , such that , so that the previous case can be applied.
Start with a triangle with the same area . (For example, if is at height from , choose arbitrarily and take at height .) Suppose . Draw a parallel to through . Let be the intersection of and the circle of radius with center . All three triangles , and have the same area. The triangles and have the same base and equal altitudes, so they are equidecomposable, say using a set of polygons . The triangles and have equal bases and , so they are equidecomposable, say using a set of polygons . Then and are equidecomposable using the intersections of and in .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Reference
[1] B. I. Agrunov and M. B. Balk, Elemenarnaja Geometrija (in Russian), (Elementary Geometry), Moscow 1966, pp. 174–175.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+