AoPS Academy
and 
. Prove that
Equality holds when 
or 
, 
, 
and 
lie on a circle 
such that 
. The tangent line to at point intersects the tangent line to at and the line 
at 
and 
. The circle and the circumcircle of triangle 
intersect again at 
. Prove that 
.
and . Prove that
Equality holds when or 
intersect its circumcircle 
at 
. The tangets to from 
determine a triangle 
, where 
is relative to and so on. Prove that lines 
are concurrent at a point 
and that belongs to the euler line of .
the centroid is 
and is the midpoint of 
. The line through parallel to 
meets 
at 
. Prove that 
if, and only if, 
.
be the intersection of 
and 
, M midpoint of .
.
, or 
cyclic.
, equivalent to isosceles trapezoid, or 
. (Note 
, 
are not parallel.)
, thus equivalent to 
.
, by definition of M. Clearly, 
if and only if 
.
be a midpoint of .If we have 
. As then 
which implies 
thus 
is cyclic . As is cyclic and 
parallel to we have 
thus 
so 
. Second case is just working backwards.
