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 
.
.
sastifying 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
.
Y by Adventure10, Mango247
Let 
be an internal point of triangle 
. The line through
parallel to
meets 
at 
, the line through parallel to
meets
at 
, and the line through parallel to meets at
. Prove that
and locate the position of in triangle when equality holds.
be an internal point of triangle 
. The line through parallel to
meets 
at 
, the line through parallel to meets
at 
, and the line through parallel to meets at
. Prove that
and locate the position of
in triangle when equality holds.
meet at 
.
the areas of 
,
,
is the length of 
,
is the area of ![\[ \frac {BL}{LC} = \frac {k_1}{k_2 + k_3} = \frac {\frac {\sqrt {x}}{\sqrt {K}}}{\frac {\sqrt {y}}{\sqrt {K}} + \frac {\sqrt {z}}{\sqrt {K}}} = \frac {\sqrt {x}}{\sqrt {y} + \sqrt {z}}\]](http://latex.artofproblemsolving.com/1/9/b/19b8bd17d69b32d074a10f6d03b3666f7a9ac95a.png)
![\[ \frac {CM}{MA} = \frac {\sqrt {z}}{\sqrt {x} + \sqrt {y}}, \frac {AN}{NB} = \frac {\sqrt {y}}{\sqrt {z} + \sqrt {x}}\]](http://latex.artofproblemsolving.com/9/6/b/96bfd30110c80a904f2cae518a2db48e85d318de.png)
![\[ (\sqrt {x} + \sqrt {y})(\sqrt {y} + \sqrt {z})(\sqrt {z} + \sqrt {x})\geq 8\sqrt {xyz},\]](http://latex.artofproblemsolving.com/9/0/0/9005e06fbac4ac0ce7505e9d0349b1a64638e925.png)
,
be the intersection of lines 
with 
by similar triangles![$ =\frac{[ABP]}{[ABC]}-1$](http://latex.artofproblemsolving.com/6/f/e/6fe7accd102186e5e6946e93fb2aa38e78a3dc78.png)
since the triangles have the same bases![$ [ABP]=x, [BCP]=y, [CAP]=z, [ABC]=x+y+z$](http://latex.artofproblemsolving.com/e/c/a/ecaf8085875ca7082696ced34c66fd5a0fe8b2de.png)
and clearing denominators we get the same inequality as above.
