I am solving a problem which requires a proof of the statement in the title. So far, I was considering the following cases:
- 4 angles are at least 90°: rectangle, the statement holds.
- 3 angles are at least 90°: by applying the law of cosine we immediately conclude that the longer diagonal is longer than all 4 sides
- 2 angles are at least 90°: we distinguish two cases: If the two large angles are opposite, we can conclude the same as in the previous case. Otherwise, we can say that the longer diagonal is longer than at least 3 sides.
- only 1 angle is at least 90°: ???
The remaining possibility is not clear to me. Can anyone suggest a justification why should the statement from the title hold even for the last case?