AoPS Academy
be a triangle, and let 
be an arbitrary point on line 
, where 
is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of .
be an equilateral triangle, 
be a point inside it, and 
be the intersections of 
with the sides of . If 
are the midpoints of 
, 
, 
, show that there is a triangle with sides 
, 
and 
.
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each 
along a great circle of .
and 
lying in parallel planes. Describe the locus of the midpoint of 
when 
varies along 
for 
.
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point . Let be a point on the arc 
(not containing ) such that 
and 
. Let the lines and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
. lies on the circumcircle of 
.
and two lines 
and 
are given in the space. For each nonnegative integer 
we denote by 
the projection of 
on 
and by 
the projection of on 
Prove that there exist two segments ![$[A'A''] \subset d_1$](http://latex.artofproblemsolving.com/8/8/6/88667ee717ddd630efd34a9c4c83ee359c4897ed.png)
and ![$[B'B''] \subset d_2$](http://latex.artofproblemsolving.com/3/7/4/374927492266c8adaed5d34e3871d250baba9c1b.png)
of length 
and a nonnegative integer such that ![$A_n \in [A'A'']$](http://latex.artofproblemsolving.com/4/8/5/4851110dd8f0bf0ee7e42239ee051da76f5fbd65.png)
and ![$B_n \in [B'B'']$](http://latex.artofproblemsolving.com/5/e/7/5e7d78dd940f5c81673b779a9b45e83f4dc05b4d.png)
for any 
opposite the vertex 
be tangent to the side at the point 
. Define the points 
on and 
on analogously, using the excircles opposite and , respectively. Suppose that the circumcentre of triangle 
lies on the circumcircle of triangle . Prove that triangle is right-angled. is great if the following holds: for any point 
on the side , if and are the feet of the perpendiculars from to the lines and , respectively, then the reflection of in the line 
lies on the circumcircle of the triangle . Prove that triangle is great if and only if 
and 
.. Let be the foot of the altitude of triangle issuing from the vertex , and let be the circumcenter of triangle . Assume that 
. Prove that 
.
,
,
such that 
,
in terms of 
and 
,
.