bonanova

Consider the quadrilateral ABCD where AC and BD are its two diagonals and DA is its shortest side. UHP215 Prove (or disprove): AB + BC + CD > AC + BD

Edit -- Sorry, I kept messing up. Yes to the first part, not sure about your formula.

It cannot be said with certainty that my triangle is equilateral.

No. It is too hard. Perhaps you could give us a hint.

Well, yes. But that can be true of scalene triangles as well. What can we say with certainty about my hypothetical triangle, [knowing only that its three largest (internal) squares have equal area] that distinguishes it from any triangle that does not meet that criterion? One possible answer, of the type sought, is that the triangle is equilateral. But it turns out that my triangle is not equilateral. So the answer takes the form, it is EITHER equilateral OR it is ___________________.

Did you get the second solution yet?

You probably have. I just don't know where to start. I think my triangular box problem is easier. But it makes sense that a puzzle is clear to its creator.

Yep, brain fart. But the conclusion holds even if xxxx is permitted.

[spoiler=Wilma needs an upgrade]Assuming they are English words having no special characters, only 358,800 words could be distinct. 26x25x24x23.

OK so I totally misread the OP, and Rainman provided the requested elegance. I read it as a challenge rather than as a question.

@k-man possibly so. The proof depends on whether the hemisphere is open or closed.

Yeah, what Rainman said. I figured my solution was illegal, and OP says something elegant is needed. But without going off-grid or rotating slightly, which gavinksong has now made explicitly unacceptable, I think it can't be done. I agree with Rainman. Nice proof.

Edit -- decided to use subscripts and the actual symbol for π (which is alt + p on macs btw) I played with LOC for a bit, but OP asked for the relationship among a b c d. I found I could not eliminate all the angles, only one of them. This is the relationship among a b c d. I didn't derive it, or use it, opting instead in the specific case of a b c = 3 4 5 for iterative solution.

That helps. OP does not specify the numbers on the die faces.

I believe that is the longest character string ever posted on BrainDen. The answer is fewer than nine.

Perhaps the OP was not worded as well as it could have been. I don't know it it matters, but re-read it now.

Actually, not.