intersect at point 
.
and 
with their common external tangents lie on a circle 
.
and 
with their common external tangents lie on a circle 
.
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?
Y by
Let 
be an cyclic quadrilateral and 
its circumscribed circle. Let 
be the intersection point from 
and 
and 
the intersection point of the tangents to at points 
and 
. Show that 
is a cyclic quadrilateral if and only if 
bisects the segment 
.
be an cyclic quadrilateral and 
its circumscribed circle. Let 
be the intersection point from 
and 
and 
the intersection point of the tangents to at points 
and 
. Show that 
is a cyclic quadrilateral if and only if 
bisects the segment 
.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:49 AM
is a symmedian of 
.
,
,
,
is isogonal to 
