is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
and incenter 
.
at 
respectively. Circumcircle of triangle 
intersects 
distinct from 
,
intersects 
distinct from 
.
points 
are concyclic.
.
is a parallelogram. Prove that 
.
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
that satisfy the equation 
.
be a circle, and let a line 
intersect 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
,
,
is not greater than 
.
.
.
Y by HWenslawski
Let 
and 
be the midpoints of the two sides of the triangle 
, 
and 
be the feet of the corresponding angle bisectors, and let 
and 
be the intersection points of the extensions of these angle bisectors with the circumscribed circle. It turned out that the lines 
, 
and 
intersect at one point. Determine the type of triangle .
(O. Karlyuchenko, A. Bashkirevich)
and 
be the midpoints of the two sides of the triangle 
, 
and 
be the feet of the corresponding angle bisectors, and let 
and 
be the intersection points of the extensions of these angle bisectors with the circumscribed circle. It turned out that the lines 
, 
and 
intersect at one point. Determine the type of triangle .(O. Karlyuchenko, A. Bashkirevich)
This post has been edited 2 times. Last edited by parmenides51, Jul 4, 2021, 11:56 PM
