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is given on the plane. Let the points 
, 
and 
be on the sides 
, 
and 
, respectively. (, and are different from the vertices 
, 
and 
) Also the points 
, 
and 
are taken such that 
is an equilateral hexagon. Suppose that the circumcenters of and 
are coincident. Then determine the least possible value of:
Note: 
area of 
such that ![\[f(x)(f(yf(x)+1))=f(x)+f(y)\]](http://latex.artofproblemsolving.com/1/0/6/1069c29ad9eeb7fe8597d7bac806006a12ff50c9.png)
for all positive reals 
and 
and also satisfies the property that ![\[\mathbb{R}^+\subseteq\frac{\text{Im}(f)}{\text{Im}(f)},\]](http://latex.artofproblemsolving.com/1/0/2/102f23eabefae24b158fbd7516b58b1e879c1b74.png)
or in other words, the set of positive reals is a subset of the set ![\[\left\{\frac{x}{y}\mid x,y\in\text{Im}(f)\right\}\]](http://latex.artofproblemsolving.com/f/d/4/fd4e9cf0bfc63603529ded7679e2811560219338.png)
and 
be two circles with respective centres 
and 
intersecting in two distinct points and such that 
is an obtuse angle. Let the circumcircle of 
intersect and respectively in points 
and 
. Let the line 
intersect in ; let the line 
intersect in . Prove that, the points 
are concyclic.
be a triangle. For a point 
of the plane, let 
be the foot of the perpendicular dropped from 
and 
are defined analogously. Find the locus of points ![\[PA' \cdot PA=PB' \cdot PB=PC' \cdot PC \]](http://latex.artofproblemsolving.com/a/f/e/afef9b0b21f8e6f241048d434f3e9df4b9b75c43.png)
