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.)
be an acute-angled, non-isosceles triangle with circumcenter 
and incenter 
,![\[
\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,
\]](http://latex.artofproblemsolving.com/7/8/1/78125759bf5c73149e4a56538e9b0c44309704a7.png)
,
,
.
,
,
be the excenters opposite to vertices 
,
,
,
,
,
be the Lemoine points of triangles 
,
,
,
,
,
all pass through a common point 
.
.
intersect at point 
.
and 
with their common external tangents lie on a circle 
.
and 
with their common external tangents lie on a circle 
.
Y by Mango247
Let 
be a triangle with an obtuse angle 
. The incircle of centered at 
is tangent to the sides 
at 
respectively. Lines 
and 
intersect 
at 
and 
respectively. Let 
be the midpoint of 
. Show that 
lie on a circle.
be a triangle with an obtuse angle 
. The incircle of centered at 
is tangent to the sides 
at 
respectively. Lines 
and 
intersect 
at 
and 
respectively. Let 
be the midpoint of 
. Show that 
lie on a circle.This post has been edited 1 time. Last edited by parmenides51, Aug 16, 2020, 7:53 PM
Reason: latex
Reason: latex
.
so 
are concyclic. Let 
.
.
.
which follows from the fact that 
and 
