be the circumcenter, symmedian point of 
.
at 
,
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
Y by Adventure10, Mango247
Let 
and 
be two points on the arcs 
and 
of the circumcircle of a triangle 
, respectively, such that 
. Show that the incenters of triangles 
and 
are equidistant from the midpoint of the arc of the circumcircle of triangle .
and 
be two points on the arcs 
and 
of the circumcircle of a triangle 
, respectively, such that 
. Show that the incenters of triangles 
and 
are equidistant from the midpoint of the arc of the circumcircle of triangle .
and 
the inceters of 
and 
,
and 
to meet the circumcircle(for the second time) of 
at 
and 
,
.
-
is found on the perpendicular bisectors of 
,
and 
and consequently 
.
and 
and 
.
is a trapezoid so 
.
so 
.
so we conclude that 
(trivial) and 
we have that 
so 
complex numbers: 
.
we get 
.
.
.
it follows 
be the incentres of triangles 
and 
respectively. Further, let 
be the midpoint of minor arcs 
.
to 
.
,
.
,
as well, as desired.![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
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and 
denote the incenters of 
and 
,
,
,
,
,
