I came across a property which I am not sure if true in general. Suppose you have a parallelogram whose vertices are v_1,v_2,v_3,v_4 in R^2. Let's say that the side [v_1,v_4] is parallel to [v_2,v_3]. Is it true that ||v_1−v_4||^2<||v_1−v_3||*||v_2−v_4||, that is, the square of one side is less than the product of the diagonals? I could also be just missing an obvious counterexample, but so far I can't prove it either.

sarahkobearab4

sarahkobearab4

Open question

2022-08-18

I came across a property which I am not sure if true in general. Suppose you have a parallelogram whose vertices are v 1 , v 2 , v 3 , v 4 R 2 . Let's say that the side [ v 1 , v 4 ] is parallel to [ v 2 , v 3 ]. Is it true that
| | v 1 v 4 | | 2 < | | v 1 v 3 | | | | v 2 v 4 | | ,
that is, the square of one side is less than the product of the diagonals? I could also be just missing an obvious counterexample, but so far I can't prove it either.

Answer & Explanation

Erika Brady

Erika Brady

Beginner2022-08-19Added 19 answers

Consider a rhombus with unit sides which is very long and thin. As it gets pointier, one diagonal tends to zero and the other to length 2, so the product of the diagonals gets very small.

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