and 
.
Where 
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
,
and 
.
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
Y by Adventure10
Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the
circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.
circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.
![$ [A'B'C'] = \frac{R^2}{4} \left ( \tan A + \tan B + \tan C \right ) = \frac{R^2}{4} \tan A \tan B \tan C$](http://latex.artofproblemsolving.com/2/9/d/29db7085780ada74c9c342e9da102467717d8cb9.png)
and ![$ [ABC]= 2R^2 \sin A \sin B \sin C$](http://latex.artofproblemsolving.com/6/7/e/67e2c7dfeca9fa587ae5c1d7b06e57fbb6d106bb.png)
.
that is equal to the well-know 
.
.
