satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
be the excircle opposite to 
.
and 
be the points where 
,
,
intersects line 
at 
.
be the midpoint of 
.
is tangent to 
,
,
and 
are the sides of a quadrilateral inscribed in a given circle. Prove that the product 
acquires its maximum when the quadrilateral is a square.
satisfy that 
.
Y by Adventure10
In acute 
and 
are the orthocentre and the circumcentre respectively.Prove that for any point 
on the segment 
we have 
where 
is the inradius for 
.
and 
are the orthocentre and the circumcentre respectively.Prove that for any point 
on the segment 
we have 
where 
is the inradius for 
.
