be an acute triangle and 
its orthocenter. Let 
such that 
.
and through 
similarly. If 
are positioned around the sides of 
,
and 
the intersection of the circumcircle of 
,
is the reflection of 
the midpoint of 
,
,
and the perpendicular through 
and 
a point on 
,
.
lie on a line.
and any 
,
are also in 
for positive integers 
and 
.
be an isosceles triangle with 
.
be a point on 
.
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.
Y by
Let 
and 
be points on a circumference 
with center 
, in that order, such that 
is perpendicular to 
; 
is their intersection. 
is the convex quadrilateral such that and are the tangency points of with segments 
and 
respectively. The perpendicular lines to 
and 
through 
and 
, respectively, intersect at 
; the perpendicular lines to and 
through and 
, respectively, intersect at 
, and the perpendicular lines to and 
through and 
, respectively, intersect at 
. 
is the circumcenter of triangle 
. Prove that 
and are collinear.
Proposed by CDMX
and 
be points on a circumference 
with center 
, in that order, such that 
is perpendicular to 
; 
is their intersection. 
is the convex quadrilateral such that and are the tangency points of with segments 
and 
respectively. The perpendicular lines to 
and 
through 
and 
, respectively, intersect at 
; the perpendicular lines to and 
through and 
, respectively, intersect at 
, and the perpendicular lines to and 
through and 
, respectively, intersect at 
. 
is the circumcenter of triangle 
. Prove that 
and are collinear.Proposed by CDMX
This post has been edited 2 times. Last edited by Justpassingby, Jan 20, 2022, 1:31 AM
Reason: Added proposer
Reason: Added proposer
.
say)
.
are colinear (and so are 
)
.
,
and 
-
.
are colinear: 
so 
and 
so 
.
to be parallel to 
.
