Wikipedia says that

The shape of a simple quadrilateral is fully determined by the lengths of its sides and one diagonal.

but I have my doubts. For example, the two quadrilaterals in this picture both have the same side lengths and the same yellow diagonal, but are not the same.

two simple quadrilaterals

Am I missing something here, or are simple quadrilaterals not actually determined (all sides and angles) by the lengths of four sides and a diagonal? If not, what about a convex quadrilateral? Would that be fully determined by those lengths?

  • $\begingroup$ Perhaps what is meant is not either of the diagonals, but (at least) one of them. $\endgroup$ – Benjamin Dickman Nov 6 '12 at 8:34
  • $\begingroup$ If by "lengths" of the sides one means the lengths in say counterclockwise order, then the lengths of the sides together with the length of a diagonal to specified vertices does determine the quadrilateral if it is convex. Your example shows the result is not true if we drop convexity. The Wikipedia article specifies simple here, and "simple" does not imply convex. So, a mistake. $\endgroup$ – André Nicolas Nov 6 '12 at 8:38

The wikipedia article specifies that the polygon is convex in almost every paragraph of this article.

It is only true for a convex polygon, you just proved that wikipedia can be wrong :).

For a convex polygon i can think of the following proof :

With 4 lengths and 1 diagonal you define 2 unique triangles (a triangle is uniquely defined by the length of his 3 sides) with one common side, and therefore one unique polygon.

  • $\begingroup$ It does specify a convex polygon in almost every paragraph, except the one I quoted from. Hence the confusion. (You wouldn't happen to know of a proof, would you?) $\endgroup$ – David Z Nov 6 '12 at 8:42
  • $\begingroup$ @DavidZaslavsky I added a very quick proof for a convex polygon, and your question prove that it's not true for not convex simple polygons. $\endgroup$ – Ricky Bobby Nov 6 '12 at 8:47
  • 5
    $\begingroup$ I fixed the wikipedia article. If we want to be pedantic, we perhaps add that the order of the side lengths matters, as well as which diagonal is given. $\endgroup$ – Harald Hanche-Olsen Nov 6 '12 at 9:12

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