Using your Head is Permitted

Let ABCD be a quadrilateral with side lengths a≥b≥c≥d and diagonal lengths e and f. Prove or disprove: e+f ≤ a+b+c.

Several readers sent in their ideas for related inequalities. In particular, Radu-Alexandru Todor sent in a whole bunch of such inequalities and related problems. Here are three of his inequalities, which you can try to prove (or disprove!) this month.

1. e+f ≤ a+b+d.
2. e+f ≤ 0.5a+1.5b+0.5c+0.5d.
3. e+f ≤ a+b+(√2-1)c+(√2-1)d.

I will stress that Radu not only wrote this month's riddle for me, but also wrote up the entire solution which I will present at the end of the month. Thanks Radu!

In a break from tradition, I will not be soliciting reader proofs this month. and there will be no list of solvers. Please only send in a solution if you think that it is so extraordinarily elegant that I may want to integrate it alongside Radu's original solution.

No List of solvers this month.