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

1. A quadrilateral can be partitioned into 2 equal-area triangles if and only if one diagonal divides equally the other diagonal.

2. It can be proved that most quadrilaterals cannot be partitioned into 3 equal-area triangles.

3. 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).]

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## 1 Answer

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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