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 
respectively. Let 
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
Let 
be a triangle, 
its incenter, and 
a point on the arc 
of the circumcircle of not containing 
. The bisector of the angle 
intesects the segment 
at 
. The bisector of the angle 
intesects the segment 
at 
. Prove that the points 
are collinear.
(Malik Talbi)
be a triangle, 
its incenter, and 
a point on the arc 
of the circumcircle of not containing 
. The bisector of the angle 
intesects the segment 
at 
. The bisector of the angle 
intesects the segment 
at 
. Prove that the points 
are collinear.(Malik Talbi)
bisects 
and 
bisects 
,
(not containing 
)
and 
meet at the midpoint 
of arc 
.
,
