[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 93
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DAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA sadistIn principle, we are already well advanced and have drawn circles on the sides of the quadrilateral. It remains to find the right starting point of one circle to start drawing from to get an exact square.
What I liked about this problem is that the resulting 4 intersecting circles represent an interesting system. It is clear that the sides of the square go through their intersection points. However, if we take any segment passing through any of these points, and complete it further by the points of its intersection with the circles (I hope you see what I mean), then this polyline will close and we obtain a rectangle inscribed into these 4 circles. Visually it turns out that the rectangle changes its proportions, keeping the corners. The square we are looking for is the only particular case of this rectangle when the sides become equal. And in the aforementioned degenerate case, the rectangle turns out to be a square and does not change its proportions when rotated.
This led me to a funny idea: we can construct two arbitrary rectangles on opposite sides of the square we are looking for, and then iteratively find it by binary division of the angle between them. :-)
But I can't think of a method of direct construction.
PS
Alexey, nice problem. But not in the sense of solving it in the class. :-)
Необходимо найти пятую точку, принадлежащую квадрату, используя свойства квадрата.
like a tip (
Can't the dosage be increased?
In principle, we are already well advanced and have drawn circles on the sides of the quadrilateral. All that remains is to find the right starting point of one circle, which is the starting point from which we must start drawing to get an exact square.
I have solved the problem without drawing circles, I am sitting, laughing, waiting for how it all ends :)))
I have to leave.
And here they are... they're handing out clues in homeopathic doses, soon they'll be shoving clues into crossword puzzles, sadists
Yes, Yuri, I've been looking for how they're connected, those rectangles. I couldn't find any.
2 TheXpert: The point of diagonal intersection probably won't help.
You can centre the side of a square, and then draw a straight line through it and one vertex of the quadrilateral. In principle, this element is the most logical: unambiguous construction is only possible in the nondegenerate case.
Oh, also: at this point the square is touched by an inscribed circle.
а дозировку нельзя увеличить ?
Properties of a square ;) -- is a rectangle with equal sides. That's it, there's no more to tell.
Свойства квадрата ;) -- это прямоугольник с равными сторонами. Все, дальше подсказывать некуда.
That's it, I'm a down-ambicil (Here's my solution:
1. What is the sum of the interior angles of the square?
2. What is the sum of angles of a quadrilateral?
3. What is the sum of angles of an expanded angle?
4. What is the sum of angles of a triangle equal to?
-
Using 4 points you can construct an infinite number of rectangles, but only one will be a square :)