and 
such that 
and 
is a prime number
be the midpoint of side 
intersect 
at 
.
is tangent to 
,
.
be a trapezoid such that 
lie on a circle with center 
,
is parallel to side 
.
and 
intersect at point 
,
.
.
,
and 
are positive integers and 
.
?
is a regular tetrahedron.
be a trapezium in which 
and 
.
be then midpoint of the diagonal 
.![$[ABCD] = n \times [CDE]$](http://latex.artofproblemsolving.com/e/9/3/e9347ba2c8e9396adfe716159e711408a3b9dc69.png)
,
?![$[t]$](http://latex.artofproblemsolving.com/8/a/8/8a86702dc60b9db916798b1115a6b6822bf828b9.png)
denotes the area of the geometrical figur
.
.
we denote the angle bisector of angle 
. Let 
and 
.
is cyclic.
,
.
and 
,
with respect to a circle 
,
,
,
Y by Adventure10
Consider a sequence of circles 
of radii 
, respectively, situated inside a triangle 
. The circle 
is tangent to 
and 
; 
is tangent to , 
, and 
; 
is tangent to , 
, and 
; 
is tangent to , , and ; etc.
(a) Prove the relation
![\[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \]](//latex.artofproblemsolving.com/4/b/5/4b58ea2f504c79401d29ff70f96fb43d1ef9fb02.png)
where
is the radius of the incircle of the triangle . Deduce the existence of a 
such that
![\[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\]](//latex.artofproblemsolving.com/4/3/2/4324c7c175cd2615abffcdf495eaba738dc8ff9b.png)
(b) Prove that the sequence of circles
is periodic.
of radii 
, respectively, situated inside a triangle 
. The circle 
is tangent to 
and 
; 
is tangent to , 
, and 
; 
is tangent to , 
, and 
; 
is tangent to , , and ; etc.(a) Prove the relation
![\[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \]](http://latex.artofproblemsolving.com/4/b/5/4b58ea2f504c79401d29ff70f96fb43d1ef9fb02.png)
where
is the radius of the incircle of the triangle . Deduce the existence of a 
such that![\[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\]](http://latex.artofproblemsolving.com/4/3/2/4324c7c175cd2615abffcdf495eaba738dc8ff9b.png)
(b) Prove that the sequence of circles
is periodic.
