AoPS Academy
be a triangle with incenter 
, and suppose the incircle is tangent to 
and 
at 
and 
. Denote by 
and 
the reflections of and over . Let 
be the intersection of 
with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
that verify the following equation for all 
:
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of . Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through tangent to 
respectively. Let be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter , centroid and orthocenter . Prove that 
is constant.
that satisfy the equation 
.
be a circle, and let a line 
intersect at two points, and 
Circles and 
are internally tangent to at points 
and 
respectively, and both are tangent to at a common point 
Similarly, circles 
and 
are externally tangent to at and respectively, and are tangent to at points and 
respectively. is 
the segment 
has a length of 
and 
find the length of segment 
be a triangle with side lengths being integers and and 
being relatively prime. Let the tangent at A to the circumcircle of meet produced at . Prove that both and 
are rational, but that neither is an integer.
be a triangle with side lengths being integers and and being relatively prime. Let the tangent at A to the circumcircle of meet produced at . Prove that both and are rational, but that neither is an integer.
and 
Prove that
.
Prove that
and 
Prove that
