is given where 
are positive reals. Prove that:
be a triangle and let 
be a circle in its plane passing through 
and 
Suppose there exist circles 
such that for 
is externally tangent to 
and passes through 
and 
where 
for all 
.
and 
have common external tangents 
and 
meeting at 
.
and 
.
through 
and 
,
is convex.
meet at point 
,
and 
.
Y by ImSh95, Rounak_iitr
Let 
be an acute scalene triangle with circumcircle 
. Let 
and 
be interior points of the sides 
and 
, respectively, such that 
is parallel to 
. Let 
be a point on such that 
is parallel to . The segments 
and 
intersect at 
. The line 
intersects at 
. Prove that 
.
Proposed by Ervin Macić, Bosnia and Herzegovina
be an acute scalene triangle with circumcircle 
. Let 
and 
be interior points of the sides 
and 
, respectively, such that 
is parallel to 
. Let 
be a point on such that 
is parallel to . The segments 
and 
intersect at 
. The line 
intersects at 
. Prove that 
.Proposed by Ervin Macić, Bosnia and Herzegovina
This post has been edited 1 time. Last edited by Lukaluce, Jul 2, 2022, 10:08 PM
and 
are isogonal in 
and 
and 
and 
,
and 
are isogonal in 
as well: note that there is an involution at 
and because of (*), this involution is a symmetry wrt the angle bisector of 
intersects 
at 
;
.
is the midpoint of 
are concurrent, therefore 
,
,
is the external bisector of 
is the internal bisector of 
,
is cyclic.
is cyclic, therefore 
