grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
be the circumcenter and orthocenter of a scalene triangle 
be the intersection point of the lines 
and 
meets the side 
.
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
,
and 
be a finite set of cells in a square grid. On each cell of 
and 
the perpendiculars in 
respectively, where 
,
and 
.
be the circumcircles of triangles 
.
and 
,
Y by
Inside triangle 
(
) there is a point 
equidistant from its three vertices. 
is the bisector of angle 
. Point 
is the midpoint of side 
, and point 
on ray
is such that 
. Point 
is the foot of the perpendicular drawn from on 
. Prove that each of the diagonals of the quadrilateral 
divides the triangle into two equal parts.
(
) there is a point 
equidistant from its three vertices. 
is the bisector of angle 
. Point 
is the midpoint of side 
, and point 
on ray
is such that 
. Point 
is the foot of the perpendicular drawn from on 
. Prove that each of the diagonals of the quadrilateral 
divides the triangle into two equal parts.This post has been edited 2 times. Last edited by parmenides51, Jan 7, 2021, 9:42 PM
