AoPS Academy
be a triangle, and let 
be a point on side 
. A line through intersects side 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point 
distinct from point 
. The lines 
and 
intersect again at 
and 
respectively.
such that 
for all real number 
and 
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let intersect 
and 
at and respectively and intersect 
and 
at and respectively .
, point 
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds. is an isosceles triangle.
be an arbitrary point inside a tetrahedron 
.![$[AOC] \cdot [BOD] \le [AOB] \cdot [COD] + [AOD]\cdot [BOC]$](http://latex.artofproblemsolving.com/a/8/6/a86d4ce3f216daebd6e01087f202d25393a2ca24.png)
.![$[XYZ]$](http://latex.artofproblemsolving.com/c/7/6/c7608a3e97e44fc5bcf0cd62a1f7dfc0a0a2e7d4.png)
stands for the area of triangle 
.)
be an arbitrary point inside a tetrahedron .. stands for the area of triangle .)
