AoPS Academy
intersect at point 
. The points of tangency of the circumcircles of triangles 
and 
with their common external tangents lie on a circle 
. The points of tangency of the circumcircles of triangles 
and 
with their common external tangents lie on a circle 
. Prove that the centers of circles and coincide.
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?
, its three diagonals 
,
,
are concurrent at 
.
, 
, 
are 
, 
, 
respectively.
and similarly for the other triangles, you will find
.
takes its minimum, 
, when all three areas are equal to , by AM-GM.
. You can draw a diagram to show this minimum is achievable (might as well let 
, 
, 
all be equilateral)
