System of equations

[ins-]

It is given:
$x_1=\frac{(a+b+b+c-y)x}{2b}$
$x_2=\frac{(a+d+d+c-y)x}{2d}$
$x_3=\frac{(b+a+a+d-z)x}{2a}$
$x_4=\frac{(b+c+c+d-z)x}{2c}$
$y^2=\frac{a+c}{b+d}((a+c)(b+d)+4bd)$
$z^2=\frac{b+d}{a+c}((a+c)(b+d)+4ac)$
$x^2=\frac{abc+bcd+cda+dab}{a+b+c+d}$
$x>x_1,x_2,x_3,x_4$
$x>a,b,c,d$
$b>x_1,d>x_2,a>x_3,c>x_4$
All variables are positive reals.
a) Can we find a dependency between $x_1,x_2,x_3,x_4$ and $x$?
b) Can we find a dependency between $x_1,x_2,x_3,x_4$?

[Bulinator]

Is this really a problem from an olympiad?
Try to use Solve[] function in Mathematica. It will exclude all the variables which can be excluded.

[ins-]

I cannot use that function. Can you tell me the result. The problem have an application in geometry it is the reason to post it. I'll be very happy to know if there is some dependency between the variables. The problem I'm trying to solve is a "Sangaku" style problem. It is the statement:

It is given a circimscribed quadrilateral ABCD. Do there exist a dependency between $r_{ABC}$, $r_{BCD}$, $r_{CDA}$, $r_{DAB}$? What if we include $r$?
Here $x_1, x_2, x_3, x_4$ and $x$ are the radii of the incircles mentioned and a,b,c,d are the respective tangents. $y$ and $z$ are the diagonals. I managed to get these equations but I failed on algebra.

[Bulinator]

ins- в сообщении #399526 писал(а):

I cannot use that function.

What is the problem? Try to use help.

ins- в сообщении #399526 писал(а):

The problem have an application in geometry it is the reason to post it.

I think the appropriate place for the thread is here.

ins- в сообщении #399526 писал(а):

The problem I'm trying to solve is a "Sangaku" style problem.

Don't know what it is.

[ins-]

Sangaku are old Japanese problems writen in their temples on a wooden tablets. They were created in the edo period from Japanese history. Some of the problems are very hard. You can take a look at the Fukagawa and Pedoe's book.

It is not usual system and I don't believe the software can handle it. It is the reason I didn't tried it.
Do you believe the software can solve all integrals possible for example?

[Bulinator]

ins- в сообщении #399533 писал(а):

Do you believe the software can solve all integrals possible for example?

Of course, no. However, I think it is better to try it first before starting to perform long and boring calculations.

[ins-]

If you have time you can try it with the software you mentioned. If it remain too long time not solved I will learn how to use such kind of software but at the moment I have no enough time. It is the reason.