AoPS Academy
be a cyclic quadrilateral, and let 
, 
, 
, and 
be the midpoints of 
, 
, 
, and 
respectively. Let 
, 
, 
and 
be the orthocenters of triangles 
, 
, 
and 
, respectively. Prove that the quadrilaterals and 
have the same area.
, let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through 
perpendicular to . Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let 
be the intersection of 
and 
. Prove that the line 
bisects segment 
.
be three positive real numbers satisfying 
Prove that
There is a very elegant proof :-D Could anyone think of it?
are equidistant from 
, so must pass through the midpoint of 
. In the same way we conclude that 
must pass through the midpoint of , while 
pass through the midpoint of 
. If 
or 
we're done: or is the diagonal we're looking for; if, on the other hand, we assume that 
and 
, from the observation above it follows that is the common midpoint of the diagonals 
, contradicting .
or 
then the conclusion follows. Then assume the contrary and note that the equality of the areas of 
and 
implies that 
or equivalently 
. Thus 
which means that 
or 
and 
are collinear. In the same fashion we get that and 
are collinear and hence and are collinear which means 
. In such a way we get a contradiction.
,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
have equal areas. Prove that the area of