AoPS Academy
be a cyclic quadrilateral. Let 
, 
, 
be the feet of the perpendiculars from 
to the lines 
, 
, 
, respectively. Show that 
if and only if the bisectors of 
and 
are concurrent with 
.
be reals such that 
. Prove that
. Let and 
be the feet of the altitudes from 
to and from 
to respectively.
and 
as the reflections of across lines and , respectively. Let 
be the circumcircle of 
. Denote by the second intersection of line 
with , and by the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
of triangle 
select point 
and on side select point 
. Segments 
and 
intersect at . Prove that quadrilateral 
is tagnential if and only if the incircles of the triangles 
and 
are tangnent.
of triangle select point and on side select point . Segments and intersect at . Prove that quadrilateral is tagnential if and only if the incircles of the triangles and are tangnent.
