,
and 
.
and 
the reflections of 
be the intersection of 
,
be the midpoint of 
and 
are perpendicular.
that verify the following equation for all 
:
.
be the feet of the altitudes from 
to 
be the circle with diameter 
.
be the intersection of 
and 
.
be the intersection of 
and 
be the lines through 
respectively. Let 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
that satisfy the equation 
.
intersect 
Circles 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
Y by Adventure10, Mango247, and 2 other users
Consider a triangle 
with 
Let 
be the midpoint of 
and 
be the intersection of the bisector of 
with . The circle through 
meets 
again at 
respectively. Let 
be a point on 
with 
. Prove that triangles 
and are similar and find the ratio of their areas.
with 
Let 
be the midpoint of 
and 
be the intersection of the bisector of 
with . The circle through 
meets 
again at 
respectively. Let 
be a point on 
with 
. Prove that triangles 
and are similar and find the ratio of their areas.
