be an acute-angled triangle, with circumcenter 
,
and orthocenter 
be a point on 
.
and 
similarly.
,
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
,
are positiv real number, such that 
prove inequality 
Y by
Define the lines 
respective to the vertices 
of the acute triangle 
as follows:
Let
be the foot of the altitude drawn from vertex 
on the side 
and let 
be the circle with diameter 
, that intersects sides 
and 
at 
and 
, respectively (
and 
); then define 
as the line passing through point perpendicular to 
. Lines 
and 
are constructed respectively.
Prove that the lines
and intersect at one point.
respective to the vertices 
of the acute triangle 
as follows:Let
be the foot of the altitude drawn from vertex 
on the side 
and let 
be the circle with diameter 
, that intersects sides 
and 
at 
and 
, respectively (
and 
); then define 
as the line passing through point perpendicular to 
. Lines 
and 
are constructed respectively.Prove that the lines
and intersect at one point.This post has been edited 1 time. Last edited by parmenides51, Sep 22, 2020, 5:47 PM
is similar to 
,
is similar to 
,
,
is cyclic, therefore 
is similar to 
