be a sequence of positive real numbers that satisfies 
and 
for every natural number 
.
for every natural number 
and 
.
be positive real numbers such that : 
. Prove that : 
.
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
Y by Adventure10
Let 
be points on the sides 
of a triangle 
, respectively. Suppose that the inradii of the triangles 
are all equal to 
. If 
and 
are the inradii of triangles 
and respectively, prove that 
.
be points on the sides 
of a triangle 
, respectively. Suppose that the inradii of the triangles 
are all equal to 
. If 
and 
are the inradii of triangles 
and respectively, prove that 
.This post has been edited 1 time. Last edited by parmenides51, Jan 19, 2020, 7:31 PM
where 
is its semiperimeter and 
and therefore 
Let 
the lengths of the common tangents from the vertices 
to their respective incircles of radius 
Then 
etc., which by adding gives us 
Now, using similar triangles we see, e.g., that 
etc., which by adding gives us 
From 
and 
we get 
which upon substitution into 
gives us 
