[/quote]
,
that satisfy the equation
with incenter 
.
at 
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
is the reflection of 
.
be the reflection of 
.
concyclic.
. Prove that: 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
is an equilateral . Let 
and 
at 
and 
at 
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
Y by Adventure10, Mango247
In convex quadrilateral 
, 
and 
. Point 
lies on segment 
and point 
lies on segment 
such that 
, , , 
lie on a circle. Point 
is such that triangles 
and 
are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point 
is such that triangles 
and 
are similar and the corresponding vertices are in the same orientation. Prove that points 
, , are collinear.
, 
and 
. Point 
lies on segment 
and point 
lies on segment 
such that 
, , , 
lie on a circle. Point 
is such that triangles 
and 
are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point 
is such that triangles 
and 
are similar and the corresponding vertices are in the same orientation. Prove that points 
, , are collinear.
be the circumcenter of 
then since 
,
,
then 
and they have the same orientation so 
and 
Similarly 
Since 
then 
so 
are collinear.
