be tangent to the side 
at the point 
.
on 
on 
and 
,
lies on the circumcircle of triangle 
be the circumcircle of triangle 
is the midpoint of 
and 
,
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be positive integers such that 
and 
.
.
.
are the midpoints of 
respectively. 
,
is symmetric to 
.
be the incenter and tangent of angles 
respectively. Prove that 
have a common point
and 
.
specifies the greatest common divisor of 
and 
.
be real numbers so that 
and 
Show that
and 
such that:
Y by Mango247
Let 
be a triangle inscribed in the circle 
and 
be center of the inscribed in circle. The lines 
and 
intersect circle again in 
and 
. Circle 
, circumscribed to 
, intersects again 
and 
in 
and 
. Show that 
and 
intersects on the circle .
be a triangle inscribed in the circle 
and 
be center of the inscribed in circle. The lines 
and 
intersect circle again in 
and 
. Circle 
, circumscribed to 
, intersects again 
and 
in 
and 
. Show that 
and 
intersects on the circle .This post has been edited 2 times. Last edited by JBMO2020, Mar 31, 2020, 4:03 PM
By Pascal theorem a
,
is collinear But, 
Hence, 
QED
)
,
,
,
,
to 
,
lies on 
.
