[/quote]
,
,
that is tangent to segment 
at 
.
-
of 
at 
.
is the midpoint of the arc 
,
.
have the same radius, then tangents to 
intersect on 
.
such that 
is a perfect square for all integers 
and 
.
such that for all 
the following holds:![\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]](http://latex.artofproblemsolving.com/a/c/a/aca098172ea7ce5663e9dd0dd10c2f1495630230.png)
be a convex hexagon, such that the triangles 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
![\[
|MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2
\]](http://latex.artofproblemsolving.com/0/7/0/070ff15920d19d474defb77ceb606b2811fa85b4.png)
at 
,
perpendicular to 
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
be the intersection of 
and 
.
bisects segment 
.
be a trapezoid with 
parallel to 
.
and 
be similar isosceles triangles with 
and 
erected externally to 
is perpendicular to 
and 
is perpendicular to 
.
and 
is its circumcenter and 
is the midpoint of the side 
.
,
intersects 
at 
.
intersects 
different from 
.
is the orthocenter of triangle 
= 
.
meets the circumcircle of 
for the second time at 
.
.
and 
,
respectively. Let 
,
,
and 
be the orthocenters of triangles 
,
,
and 
,
have the same area.
.
and 
intersect the circumcircle of 
and 
,
and 
and 
are parallelograms (with 
,
)
and 
.
,
![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](http://latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then 
,
.
at 
parallel to 
meets 
.
is constant.
Y by Adventure10, Mango247
If 
is the cevian triangle of 
, and 
are the midpoints of its sides 
, then the lines 
concur at the isotomcomplement 
of with respect to triangle 
.
P/S: The Isotomcomplement of WRT 
is the Complement of the Isotomic conjugate of .
is the cevian triangle of 
, and 
are the midpoints of its sides 
, then the lines 
concur at the isotomcomplement 
of with respect to triangle 
.P/S: The Isotomcomplement
of WRT 
is the Complement of the Isotomic conjugate of .This post has been edited 1 time. Last edited by Vanescralet, Jan 7, 2017, 4:00 AM
Reason: Adding P/S.
Reason: Adding P/S.
be the isotomic conjugate of 
WRT 
and 
be the cevian triangle of 
Let 
be the reflection of 
respectively in the midpoint of 
Clearly, 
is parallel to 
respectively, so by Pappus's theorem for 
and 
we get 
are concurrent 
hence we conclude that 
