with Incenter 
and A-Excenter 
,
to 
as its circumcenter, and 
as its orthocenter. The line 
and 
intersect the circumcircle of 
and 
be the circumcenters of triangle 
and 
respectively. If 
are not collinear, prove that 
intersects the midpoint of segment 
.
,
and 
and 
on the line 
such that 
and 
.
be the intersection of lines 
and 
.
and 
as the intersections of 
and 
with lines 
and 
,
is the intersection of 
and 
.
be a real number. Hephaestus and Poseidon play a turn-based game on an infinite grid of unit squares. Before the game starts, Poseidon chooses a finite number of cells to be flooded. Hephaestus is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop*.
after his 
t
can Hephaestus guarantee victory in a finite number of turns no matter how Poseidon chooses the initial cells to flood?
,
,
.
,
.
Y by Adventure10, Mango247
It is known that 
is an acute triangle. Let 
be the foott of the perpendicular from 
to 
, and 
, 
two distinct points on 
. The feet of the perpendiculars from to 
and 
are 
and 
, respectively. Show that if 
is a parallelogram then 
is isosceles.
is an acute triangle. Let 
be the foott of the perpendicular from 
to 
, and 
, 
two distinct points on 
. The feet of the perpendiculars from to 
and 
are 
and 
, respectively. Show that if 
is a parallelogram then 
is isosceles.
we get 
,
,
,
,
,
