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where they are integers and 
, where 
are in a gp.
board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a 
square to another midpoint of the vertical segment, according to the rules:
When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from 
to 
);
When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from 
to );
When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from to 
).
board.
be an incenter of triangle 
and let incircle touch sides 
and 
in points 
and 
, respectively. Lines 
and 
intersect line 
in points 
and 
, respectively. Prove that points , , 
and 
are concyclic
be an incenter of triangle and let incircle touch sides and in points and , respectively. Lines and intersect line in points and , respectively. Prove that points , , and are concyclic
?
. Prove that
so 
is cyclic. Hence 
and similarly 
,
is the Thales circle over 
)