be fixed points on this order on a line. Let 
be a variable circle through 
and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
.
and 
be the other points of intersection of 
and 
with 
passes through a fixed point independent of 
be a convex quadrilateral with 
.
be the point of intersection of 
and let 
.
.
and 
be the projections of 
.
meets again 
.
is the point of intersection of 
and 
intersect on the line 
.
be the circumcenter of this triangle 
,
,
,
,
,
the midpoints of these altitudes 
,
,
,
,
at the points 
,
,
,
and 
are concurrent. (When the point 
.
When does equality hold?
Y by Adventure10, mathematicsy
Let 
be an acute-angled triangle inscribed in a circle 
. It is given that the tangent from 
to the circle meets the line 
at point 
. Let 
be the midpoint of the line segment 
and 
be the second intersection point of the circle with the line 
. The line 
meets again the circle at point 
different from .
Prove that the lines and 
are parallel.
be an acute-angled triangle inscribed in a circle 
. It is given that the tangent from 
to the circle meets the line 
at point 
. Let 
be the midpoint of the line segment 
and 
be the second intersection point of the circle with the line 
. The line 
meets again the circle at point 
different from .Prove that the lines
and 
are parallel.
.
passing through 
.
is tangent to 
since 
are concyclic
,
will transform into the circle circumscribed to 
.
we get that the line 
and since 
then 
so 
then triangles 
and 
are similar 
Cyclic quadrilateral)
