[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 98
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Очевидно, построение не работает, когда диагонали перпендикулярны (четырехугольник - не квадрат) - те же АС и ВЕ.
In this case we construct a perpendicular at point E (or D, which is the same thing). There can only be one such case when the diagonals are perpendicular to the sides of the square.
Without proof ("I swear by my mother!" :) ).
Didn't let me bother :)
да, паралельная - лишнее, сразу перпендикуляр к AC можно.
шо делать, если AC перпендикулярно BD осталось решить.
Do nothing. There's a degenerate case described in a lot of posts.
шо делать, если AC перпендикулярно BD осталось решить.
Then it looks like any rectangle by these points would be a square, since the distances between the faces are equal - it's a rhombus, plus the right angles by construction.
Такой случай может быть только один -- когда диагонали перпендикулярны сторонам квадрата.
Без доказательства ("мамой клянусь!" :) ).
Have you thought it over? // Or is Mum getting in the way a lot? ;-)
Firstly, if the diagonals are perpendicular, they are equal. It cannot be otherwise. Hence, after the failure of Swan's construction (points D and E coincided) we can immediately build a square with sides parallel to the diagonals of the quadrilateral. Another thing is whether it is the only one.
P.S. Candid has already solved everything. The congenital case is when the diagonals of the quadrilateral are perpendicular.
:)
Хорошо подумал?
Yes. I've long been more of a programmer than a mathematician, and I've never liked rigorous proofs, because sometimes the obvious things are proved through the ugly *ass.
Again -- a quadrilateral is not a square.
Firstly, if the diagonals are perpendicular, they are equal. It cannot be otherwise. Consequently, after the failure of Swan'
s construction (points D and E coincided) you can immediately construct a square with sides parallel to the diagonals of the quadrilateral. Another matter is whether it is unique.Apparently not either.
Повторюсь -- 4-угольник -- не квадрат.
Then I agree.
Другое дело - единственен ли он?
If the 4-corner is not a square - then it is the only one. Otherwise, see above.
Видимо, тоже нет.
So far, something is not in sight.