I did not want to publish this problem and I wanted to solve it myself, but I was tired of dealing with this difficult problem. This problem, in a way, tells us that we need some theorems that start from areas to calculate lengths. There are many theorems in the opposite direction, but I hardly see any theorems that start from areas. The lengths are calculated
The problem first occurred to me four days ago
the problem:
We have the Descarte coordinate system and two perpendicular straight lines intersecting at a point that does not belong to the coordinate axes and does not match the coordinate axes, and they limit the Areas $A,B,C$ shown in the above figure. If the data are $A,B,C$ values, how Can we find out the coordinates of their intersection $P$? (Suppose we know at what level of the Cartesian level the point $P$) there are several ways I tried to deal with the problem but I haven't come up with a final solution yet, I will put some ideas on the problem:
perhaps the first idea and the easiest idea is to know that the two triangles are in the area. The same is equal to the square of the similarity, this idea gave me a large number of proportions that can be formulated, and certainly the theorem of Pythagoras also appears with these existing triangles and square proportions, but this method was a great confusion, in front of me a large number Of the equations and distractions and I don't know what to do.
A second idea to proceed with the solution: we know that one of the definitions of the hiperbola is that it represents the geometrical place of a moving straight line that confines a fixed space by its intersection with two fixed intersections.
$f(x):y=\frac{-(A+C)}{2x}$ , $g(x):y=\frac{B}{2x}$
$f'(x)=\frac{A+C}{2x^2}$ , $g'(x)=\frac{-B}{2x^2}$
$\frac{A+C}{2{x_1}^2}\cdot\frac{-B}{2{x_2}^2}=-1$
$x_2=\frac{\sqrt{B(A+C)}}{2x_1}$
$Δ_1:y+\frac{A+C}{2x_1}=\frac{A+C}{2{x_1}^2}\cdot(x-x_1)$
$Δ_2:y-\frac{B}{2x_2}=\frac{-2{x_1}^2}{A+C}\cdot(x-x_2)$
But I stopped here and couldn't finish
I suppose we can continue by finding the equation of the area enclosed between $Δ_1,Δ_2$ and the two coordinate axes and then finding when this equation equals $A$
A third way to deal with the problem is to calculate the lengths in the figure in terms of an optional major length and then deal with that length separately
$PS=a$
$OS=\frac{a\sqrt{A+C}}{\sqrt{A+B}}$
$OT=\frac{2B\sqrt{A+B}}{a\sqrt{B}}$
$PT=\frac{2(A+B)}{a}$
$OV=\frac{a\sqrt{B}}{\sqrt{A+B}}$
All of these methods may give some hope and insight to solve this specific problem, but I think that the problem is that the problem is in the lack of theorems that start from the Areas and give a few lengths, and we must start making and collecting them. The lengths are based on given areas.
Thank you