be tangent to the side 
at the point 
.
on 
and 
on 
analogously, using the excircles opposite 
and 
,
lies on the circumcircle of triangle 
be the circumcircle of triangle 
is the midpoint of 
be the second intersection of 
and 
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be positive integers such that 
and 
.
.
Y by Adventure10
Consider triangles 
in space.
(a) What condition must the angles
of 
fulfill in order that there is a point 
in space such that 
are right angles?
(b) Let
be the longest of the edges 
and let 
be the longest altitude of . Show that 
.
in space.(a) What condition must the angles
of 
fulfill in order that there is a point 
in space such that 
are right angles?(b) Let
be the longest of the edges 
and let 
be the longest altitude of . Show that 
.
.
.
.![$$h=\frac{2[ABC]}{BC}=\frac{1}{4}\cdot\frac{2\sqrt{-(2x^4+2y^4+2z^4+2x^2y^2+2y^2z^2+2z^2x^2)+2(3(x^2y^2+y^2z^2+z^2x^2)+x^4+y^4+z^4)}}{\sqrt{y^2+z^2}}=\frac{\sqrt{x^2y^2+y^2z^2+z^2x^2}}{\sqrt{y^2+z^2}}.$$](http://latex.artofproblemsolving.com/7/0/0/7008b728573d3dec529e356fc26cc234a14e4395.png)
The left inequality is
which is true because
The right inequality is
which is obviously true.
