be the circumcircle of triangle 
and 
be the second intersection of 
and 
,
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be positive integers that satisfy the inequality![\[
\frac{a}{b} < \frac{x}{y} < \frac{c}{d}
\]](http://latex.artofproblemsolving.com/8/d/9/8d982d52e655546ac29039fa6e440e68821bfe7f.png)
with 
.
,
.
Y by Adventure10, Mango247
A triangle 
and spheres are given in space 
and 
, each of which passes through points 
and 
. For points 
spheres not lying in the plane of triangle are drawn lines 
and 
, intersecting the sphere for the second time at points 
and 
, respectively. Prove that the planes passing through points 
and , touch a fixed sphere or pass through a fixed point.
and spheres are given in space 
and 
, each of which passes through points 
and 
. For points 
spheres not lying in the plane of triangle are drawn lines 
and 
, intersecting the sphere for the second time at points 
and 
, respectively. Prove that the planes passing through points 
and , touch a fixed sphere or pass through a fixed point.
through 
connecting centers of the spheres. Let it intersect the respective spheres with circles 
and intersect the circles 
at points 
and 
respectively. Because the circles 
are inverse with respect to 
lie on a line and so do 
.
remains constant while 
thus the distance from the center 
of the sphere 
is equal to 
and remains constant where 
stands for the radius of 
the lines 
with center 
remains constant thus the sphere 
formed by the rotation of 
