[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 129
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OK, the third one has been dealt with. And on the two sides and the bisector between, I hope you can?
ОК, с третьей разделались. А по двум сторонам и биссектрисе между, надеюсь, можно?
my head is already broken:))))
ОК, с третьей разделались. А по двум сторонам и биссектрисе между, надеюсь, можно?
Yes, a bit more complicated than the first two.
There is a similar problemhere:
1.4.05. В треугольнике известны длины двух его сторон и биссектриса угла между ними. Найти длину третьей стороны.
The idea is that ours should be solvable as well.
Тут есть похожая задача:
По идее должна быть решаема и наша.
This problem is not a construction problem. The missing side c is determined from the ratio
l=sqrt(ab(a+b+c)(a+b-c))/(a+b)
It does not follow from the unambiguity of the answer that it is possible to construct:)
And here I found what I was looking for, though without a solution. Looks like my intuition failed me:)))
169. Construct a triangle knowing its two sides and the bisector of the angle enclosed between them.
Тут есть похожая задача:
По идее должна быть решаема и наша.
This problem is solved quite easily by the already mentioned property of dividing the third side into segments proportional to the original sides.
But I would solve it algebraically, geometrically it reduces to ours.
And ours is solvable, I think. But I haven't solved it yet. :)
By the way, I made an observation: for any two non-equal segments there is always a triangle with two sides equal to the original segments and the bisector of the angle between them equal to the smaller of the two original segments. Nice.
// Only how to at least build it... ?-) Seems to be a special case, and I can't even get it right yet.
(a+b)^2 * (1 - l^2/(ab) ) = c^2
The side c is constructible, bastard. But I don't dare to use such a formula, and it's not nice either.
It is enough to construct a right-angled triangle with hypotenuse (a+b) and cathetus l*(a+b)/sqrt(ab). The hypotenuse is easy to build, but the cathetus is a bit more complicated.