[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 175
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Вот:
And it might not get built at all
It can. As with many geometric constructions, the construction itself must determine the area of constructability :) Remember the four-point square problem?
About the bisector. I don't know if this solution repeats what TheExpert drew, but the main thing is that it repeats my reasoning:))
First, we try to determine the geometric location of points that are ends of bisectors of all possible triangles with given sides a and b.
Let us represent our triangle in the Cartesian coordinate system
We consider the angle ACB=w as a modifiable parameter. The coordinates of vertices of the triangle are shown in the figure, it is also mentioned that the bisector divides the opposite side in proportion to the other two sides.
Let us find the coordinates of point K:
x = b*cos(w) +(a-b*cos(w))*b/(a+b) = ab/(a+b)*(1+cos(w))
y = ab/(a+b)*sin(w)
If we denote by r = ab/(a+b), we obtain
x = r*(1+cos(w))
y = r*sin(w)
Excluding the parameter w, we arrive at the following:
cos(w) = x/r-1
sin(w)=y/r, 0<w<pi
(x/r-1)^2+(y/r)^2=1
(x-r)^2+y^2=r^2, y>0
Obviously, we got the equation of the semicircle above the abscissa axis with centre at(r,0) and radius r, which is the required geometric place.
Now it is not difficult to do the construction as well. First construct a segment of length r:
Then we draw a segment CB=a, mark the segment CO=r on it. Then construct arcs of radius r centered at O, and of radius l (given length of bisector) centered at C, the point of intersection is point K (end of bisector). Draw line BK, construct an arc with centre in point C and radius b, at their intersection we have point A. The triangle is constructed.
А ведь оно может и вообще непостроиться
Right
insert the compasses into the point
stretch the leg of the compass to the furthest possible point on the circle and see if the straight line fits into the circle of the compass
Electronics question: why is this thing needed?
Fundamental, alsu. I'll take a closer look later.
What do you draw in so well?
Вопрос из области электроники
or electricians ?Фундаментально, alsu. Чуть попозже гляну посерьезнее.
А в чем ты так здорово рисуешь?
You won't believe it, in pint:)))
If I had come across such a problem at the Olympiad, I would probably have solved it that way. It is a pity that there were few construction problems at our Olympiads
or electricians ?
>>Somebody asked for a simpler way :)
Просили по проще :)
looks like an insulator