is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
and incenter 
.
at 
respectively. Circumcircle of triangle 
intersects 
,
intersects 
.
points 
are concyclic.
.
is a parallelogram. Prove that 
.
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
that satisfy the equation 
.
be a circle, and let a line 
intersect 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
.
.
Y by
A triangle 
with 
is given. The point 
is the center of an excircle touching the line segment 
at 
. The point 
is the center of an excircle touching the line segment 
at 
. The circumcircle of the triangle 
intersects 
and 
again at 
and 
respectively. The line perpendicular to 
at intersects the side at 
. The line perpendicular to 
at intersects the side at 
. Prove that 
and 
are parallel.
with 
is given. The point 
is the center of an excircle touching the line segment 
at 
. The point 
is the center of an excircle touching the line segment 
at 
. The circumcircle of the triangle 
intersects 
and 
again at 
and 
respectively. The line perpendicular to 
at intersects the side at 
. The line perpendicular to 
at intersects the side at 
. Prove that 
and 
are parallel.This post has been edited 1 time. Last edited by parmenides51, Nov 18, 2021, 9:14 PM
