such that:
be an acute scalene triangle 
be the orthogonal projection of 
on 
,
and 
are the midpoints of 
and 
respectively. Let 
and 
are points on the minor arcs 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
.
,
,
,
for all 
,
Y by Adventure10 and 1 other user
In tetrahedron 
let 
and 
be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that
![\[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]](//latex.artofproblemsolving.com/c/b/0/cb0ab14625e4ecf53de1d790a32e994cf4973663.png)
let 
and 
be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that![\[\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.\]](http://latex.artofproblemsolving.com/c/b/0/cb0ab14625e4ecf53de1d790a32e994cf4973663.png)
This post has been edited 1 time. Last edited by Amir Hossein, Apr 19, 2011, 2:14 AM
Reason: Edited.
Reason: Edited.
,
,
,
,
for heights 
,
,
,
respectively.
.
,
,
,
.
.
will be 
,
is the foot of the perpendicular from 
,
is the foot of the perpendicular from 
.
,
and 
.
,
,
are altitudes to triangles 
,
,
respectively.
.
,
,
.
.
