prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
,
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
Y by Adventure10, Mango247
A tetrahedron 
with acute-angled faces is inscribed in a sphere with center 
. A line passing through perpendicular to plane 
crosses the sphere at point 
that lies on the opposide side of plane than point 
. Line 
crosses plane in point 
that lies inside the triangle . Prove, that if 
, then 
.
with acute-angled faces is inscribed in a sphere with center 
. A line passing through perpendicular to plane 
crosses the sphere at point 
that lies on the opposide side of plane than point 
. Line 
crosses plane in point 
that lies inside the triangle . Prove, that if 
, then 
.This post has been edited 1 time. Last edited by j___d, Jul 27, 2016, 10:26 PM
.
.
is equivalent to that the points 
lie on a circle. Note that the trihedrons 
and 
are equal (the planar angles 
and 
are equal, plane angles 
and 
are equal, and also the dihedrons 
and 
are equal). It follows that the angles 
and 
are equal too. On the other hand we have 
so the triangles 
are similar. Hence 
,
