[/quote]
,
be a sequence defined by 
.
if 
.
that satisfy 
be reals such that 
P
,
,
is __
satisfying 
be a triangle and 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
,
is obtuse
is acute
,
is semi perimeter
,![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
satisfy that: 
and 
.
, such that 
Prove that :
equality cases : ?
and 
Prove that
.
is in the interior of ![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.
Y by
Given a segment 
and an equilateral triangle with side 
. Construct (with a compass and ruler) inside this triangle an equilateral triangle with side a such that so that the vertices of the outer triangle lay on the extensions of the sides of the inner.
and an equilateral triangle with side 
. Construct (with a compass and ruler) inside this triangle an equilateral triangle with side a such that so that the vertices of the outer triangle lay on the extensions of the sides of the inner.This post has been edited 1 time. Last edited by parmenides51, Jun 24, 2021, 7:29 AM
so that the inner (given with side 
this solves the given question as well
