AoPS Academy
. 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.
be a triangle whose incircle is tangent to 
, 
, 
at 
, respectively. Let 
and 
be the midpoints of 
and 
, respectively. Suppose that points 
and 
lie on 
so that 
and 
. Prove that lines 
and 
intersect on the circumcircle of 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
be a triangle with incenter 
. Let be the point of intersection between the incircle and the side , the points and are in the rays 
and 
, respectively, such that 
and 
. Prove that 
.
, then the problem is clearly true. Assume otherwise. Let 
be the second intersection of 
with line (otherwise known as the foot of the -external bisector). This is our key point:
and 
.
(using directed angles), and 
by using the fact that the original information implies 
.
and 
. However, both fractions on the RHS are equal due to the external angle bisector theorem (similar to the internal angle bisector theorem). Thus , as desired.
be a triangle with incenter . Let be the point of intersection between the incircle and the side , the points is on the ray IB . Prove that: if and only if 
.
