AoPS Academy
![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
is inscribed in circle 
. The interior angle bisector of angle 
intersects side 
and at 
and 
(other than ), respectively. Let 
be the midpoint of side . The circumcircle of triangle 
intersects sides 
and 
again at 
and 
(other than ), respectively. Let 
be the midpoint of segment 
, and let 
be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
be a acute triangle with altitudes 
. is the orthocenter of . The angle bisectors of 
intersect at 
, respectively. The circle 
with diameter intersects segment 
at 
. Show that is tangent to 
.
with real coefficents, that satisfies 
for all real numbers 
the unit circle, so 
. Suppose line 
cuts at points 
and 
.
, and similarly 
.
.
, so, by triangle inequality, we have 
, and we're done.
