Welcome to MathOverflow

Question

3

A quadrilateral can be partitioned into 2 equal-area triangles if and only if one diagonal divides equally the other diagonal.
It can be proved that most quadrilaterals cannot be partitioned into 3 equal-area triangles.
Conjecture: For any natural number $n$, there exists a quadrilateral that cannot be partitioned into $n$ equal-area triangles.

How can one prove this?

[Original question by bo.gu (MO user20491).]

edited Aug 20 at 3:51

Andres Caicedo
14.4k●1●42●79

asked Aug 20 at 2:31

bo.gu
101●4

Igor Rivin · Accepted Answer · 2012-08-20 02:37:40Z UTC

7

See this Wikipedia article.

answered Aug 20 at 2:37

Igor Rivin
31.2k●3●37●71

1

thank you very much – bo.gu Aug 20 at 2:41

One result mentioned in that article settles the conjecture. Trapezoids with a transcendental ratio between parallel sides have no equidissection. – Douglas Zare Aug 20 at 5:56

Remarkably there is no known trapezoid where the ratio of the parallel sides is algebraic and a proof of non-equidissectability has been found. (Though I suspect that when an equidissection exists this ratio satisfies an equation over the rationals for which every root has positive real part. So for example when the ratio is root 2, then as Stein conjectured, there is no equidissection). – paul Monsky Aug 21 at 11:23

How to partition a quadrilateral into a finite number of equal-area triangles

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.