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 
is harmonic. Project it onto line 
to get that 
. But since is the midpoint of , 
. So, and are parallel. However, all these cross ratios come from PoP relations only, so appreciate PoP 
. You could solve for 
by using the chord intersecton formula like this: 
and solve for , Then use the complex equation of a chord 
such that lies on it (
) and after solving for 
, show that the midpoint of , theb center of and 
are collinear.
and 
Prove that
.
Prove that
and 
Prove that
and 
intersects 
.
intersects 
.
is tangent to 
.
as desired.