with Incenter 
and A-Excenter 
,
be the foot of perpendicular from 
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 
,
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 Mango247
You are given an acute-angled triangle 
. The rectangle 
has vertices 
and 
, respectively, on the extensions of sides 
and 
beyond the point 
, 
and 
on side 
. Let 
be the median of triangle , 
be the midpoint of its altitude dtawn from the vertex , 
be the intersection point of the diagonals of the rectangle. Find the angle 
.
. The rectangle 
has vertices 
and 
, respectively, on the extensions of sides 
and 
beyond the point 
, 
and 
on side 
. Let 
be the median of triangle , 
be the midpoint of its altitude dtawn from the vertex , 
be the intersection point of the diagonals of the rectangle. Find the angle 
.This post has been edited 1 time. Last edited by parmenides51, Feb 20, 2021, 7:57 AM
