AoPS Academy
be real numbers satisfying
. Prove that:
be an acute triangle with 
, and let 
be the projection from 
onto 
. Let 
be a point on the extension of 
past such that 
. Let 
be on the perpendicular bisector of 
such that and 
are on the same side of and![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across 
and 
be 
and 
, respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and 
, respectively. Let 
and 
intersect 
and 
at 
and 
. Let 
intersect at 
. Prove that 
.
be an acute triangle inscribed in a circle 
with orthocenter 
and altitude . The line passing through perpendicular to 
intersects at . The perpendicular bisector of intersects 
at . Let 
intersect at . Let be the reflection of 
across 
. The circumcircle of triangle 
intersects at 
different from 
. Prove that 
and 
intersect on the circumcircle of triangle 
.
and 
two circles such that 
a point on and 
on such that 
is the center of and 
prove is the midpoint of arc 
. then 
are collinear and 
is cyclic 
is cyclic thus 
.
it's the easiest problem.
meet 
again at point 
. It is clear that points 
are collinear. Moreover, 
, therefore, the points 
are concyclic, say, on a circle 
. Since is the second intersection point of and , other than 
, we see that is the centre of a spiral similarity sending 
to 
. Since 
, we see that 
is the midpoint of arc 
.
and 
and 
in the other circle 
is cyclic but it is obvius because 
