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?
,
,
.
,
.
and 
Prove that:
Approach: I'm trying to factor 
to get 
.
Y by Adventure10, Mango247
Triangle 
is inscribed in circle 
. Tangent 
is drawn from 
, 
is on ray 
, 
is on ray 
. Line 
(
) intersects circle at 
, 
, and intersect 
and 
at 
and 
respectively. Prove that if 
, then 
.
is inscribed in circle 
. Tangent 
is drawn from 
, 
is on ray 
, 
is on ray 
. Line 
(
) intersects circle at 
, 
, and intersect 
and 
at 
and 
respectively. Prove that if 
, then 
.This post has been edited 2 times. Last edited by shobber, Jul 14, 2006, 9:28 AM
lies on the perpendicular bisector of the segment RS. Let (O') be the circumcircle of the triangle 
the circumcenter O' also lies on the perpendicular bisector of RS. The circumcircles (O), (O') intersect at the common vertex A and at another point A'. 
(both perpendicular to OO'). Quadrilateral ARSA' is a cyclic trapezoid, isosceles. Hence 
Quadrilateral ABCA' is cyclic, hence 
It follows that 
and the quadrilateral UCSA' is cyclic. Let a tangent to the circle (O) at A' meet the line PU at a point U'. Since 
and the quadrilateral U'SCA' is also cyclic. But the circumcircle of the triangle 
intersects the line 
at the point S and at one other point, therefore 
are identical. The points Q, T and the points A, A' are symmetrical with respect to OO' (the perpendicular bisector of RS). Since AP is a tangent to (O) at A and A'U a tangent to (O) at A', the points P, U are also symmetrical with respect to OO' hence PQ = UT.
