[/quote]
,
and 
Find the minimum 
and 
be the incircle and circumcircle of the acute triangle 
,
so that all of its sides are tangent to 
,
are both on 
.
and 
,
and 
,
and 
intersects on 
with Incenter 
and A-Excenter 
,
be the foot of perpendicular from 
to 
for which there exist positive integers 
,
,
such that the number
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 
at points 
,
and 
on the line 
such that 
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?
,
,
.
,
.
Prove that:
Approach: I'm trying to factor 
to get 
.
.
of triangle 
.
with diameter 
cuts 
.
is the symmedian line of triangle 
such that 
and 
,
.
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
be positive integers. Let 
,
be three roots of the polynomial 
.
,
,
,
,
,
,
.
Y by Adventure10, Mango247
Let 
be an acute-angled triangle. The lines 
, 
and 
are constructed through the vertices 
, 
and 
respectively according the following prescription: Let 
be the foot of the altitude drawn from the vertex to the side 
; let 
be the circle with diameter 
; let meet the sides 
and 
at 
and 
respectively, where and are distinct from ; then let be the line through perpendicular to 
. The lines and are constructed similarly. Prove that the lines , and are concurrent.
be an acute-angled triangle. The lines 
, 
and 
are constructed through the vertices 
, 
and 
respectively according the following prescription: Let 
be the foot of the altitude drawn from the vertex to the side 
; let 
be the circle with diameter 
; let meet the sides 
and 
at 
and 
respectively, where and are distinct from ; then let be the line through perpendicular to 
. The lines and are constructed similarly. Prove that the lines , and are concurrent.This post has been edited 1 time. Last edited by orl, Sep 12, 2008, 11:52 PM
,
and 
are the isogonal conjugate lines of the altitudes of the triangle. Hence they concur at the isogonal conjugate point of the orthocentre, which is the circumcentre.
,
.
.
.
Hence, 
Therefore, 
so 
,
We can similarly show that 
and 
