such that 
and 
are coprime if and only if 
?
be an acute triangle with 
,
be the projection from 
onto 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
Y by Adventure10, Mango247
A plane 
passes through a vertex of a cube so that the three edges at the vertex make equal angles with . Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto . There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane .
passes through a vertex of a cube so that the three edges at the vertex make equal angles with . Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto . There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane .
