The Lemon Fool

9873210 wrote:Math problems do not have a spirit, they have explicit stated conditions.

It's not cheating to violate assumption. Finding loopholes is a major source of advancement.

Ok, well in that case, nowhere was it stated that this piece of A4 was held in portrait mode -- although some folks may have made that assumption. Indeed, in the original posting there was no information given about the height or the aspect ratio of the paper at all.

So, if you hold it in landscape mode then the shortest crease is made by simply folding the paper in half.

mc2fool wrote:I don't know if that makes it a "better" solution or not, I'll leave that to the maths boffs to argue, but there is an assertion in their solution that I don't quite see, being FH/EH = GA/EA. Could someone enlighten me on the why of that, please?

* Here again: https://datagenetics.com/blog/january22018/index.html

∠AEG + ∠GEH = 90°
∠GEH + ∠HEF = 90°

hence
∠AEG = 90°- ∠GEH = ∠HEF

FH/EH and GA/EA are then the tangents of equal angles so they are also equal.

Stating it slightly differently ∠AEG = ∠HEF and ∠EAG = ∠EHF = 90° which is sufficient to establish that △AEG and △HEF are similar, and the ratios follow. Somebody could put this in a form that would satisfy Euclid, but over 2000 years of improvements leave me unable to remember the exact details (and I'd like to think that if somebody showed Euclid the 2000 years of improvements, he would jump at them.)

Hope the unicode comes through with angle ∠, degree ° and triangle △

9873210 wrote:Hope the unicode comes through with angle ∠, degree ° and triangle △

Yep, and thanks.