AoPS Academy
be positive integers and let 
be a non-increasing list of positive real numbers. Prove that there exists 
sets 
which partition the set 
such that 
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
, 
, and 
, are marked on the plane. You have a pencil and a double-edged ruler of width 
. Using them, you may perform the following operations:
and 
are marked, draw the line connecting and .
and 
are drawn, mark the intersection of and .
is drawn, draw a line parallel to that is at distance away from (note that two such lines may be drawn).
using these operations.
with ratio 
maps to the midline 
. This maps 
to the circumcenter of 
where 
is the midpoint of 
. But it is well-known that the circumcenter we are looking for is the nine-point center, so done.
, respectively, and 
, we can easily find that the intersection of all of these lines is 
.
, etc. are parallelograms, thus we are done.
is the reflection of 
over 
, which forms (parallelogram) 
concur at 
which is the nine point circle WRT 
distinct from its vertices and from the circumcenter 
.
and 
respectively the circumcenters of triangles 
and 
.
and 
are concurrent.
