.
are positive real numbers satisfying the condition 
,![\[
P = \frac{1}{1+x^2} + \frac{1}{1+y^2} + \frac{z}{1+z^2}
\]](http://latex.artofproblemsolving.com/4/e/6/4e6a6e3d1c959f8d16f08f7a8f842c20749fcef0.png)
Y by Adventure10, Rounak_iitr
In a square 
of diagonals 
and 
, we call 
at the center of the square. A square 
is constructed with sides parallel to those of with 
in segment 
in segment 
in segment 
in segment 
. If area of equals two times the area of , and 
is the midpoint of the 
side, calculate the measure of the angle 
.
of diagonals 
and 
, we call 
at the center of the square. A square 
is constructed with sides parallel to those of with 
in segment 
in segment 
in segment 
in segment 
. If area of equals two times the area of , and 
is the midpoint of the 
side, calculate the measure of the angle 
.
.
.
,
.
gives 
.
.![[asy]
import geometry;
point A,B,C,D,P,Q,R,S,M,O; A=(0,0); B=(0,4); C=(4,4); D=(4,0); P=(0.58578,0.58578); Q=(0.58578,3.41421); R=(3.41421,3.41421); S=(3.41421,0.58578); M=(0,2); O=(2,2);
draw(A--B--C--D--A);draw(P--Q--R--S--P);draw(A--O--D);draw(B--O--C);draw(Q--M--P);draw(M--O);
label("A",A,SW);label("B",B,NW);label("C",C,NE);label("D",D,SE);label("Q",Q,N);label("R",R,N);label("M",M,W);
[/asy]](http://latex.artofproblemsolving.com/6/9/4/694a3eece48a44835a3b3cc89a932cd362a7a86a.png)
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then 
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