such that for all positive integer 
,
Where 
denotes the number of divisors of 
.
.
,
such that for all positive integer 
.
then![\[\sum |\pm z_1\pm z_2\pm z_3|^2=2^3\sum_{i=1}^3|z_k|^2.\]](http://latex.artofproblemsolving.com/6/d/1/6d1cd28a47704aa0cac2a6c6b572816fc1db790a.png)
Generalize the problem.
and 
be isogonal conjugates with respect to 
.
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
.
be an equilateral triangle, 
be the intersections of 
with the sides of 
are the midpoints of 
,
,
,
and 
.
Y by Dadgarnia, little-fermat, Adventure10, Mango247
is a circumscribed and inscribed quadrilateral. 
is the circumcenter of the quadrilateral. 
and 
are the intersections of 
, 
and 
respectively. 
and 
are points on 
and 
such that 
and 
. 
and 
are points on 
and 
such that 
and 
. 
is the midpoint of arc 
of 
which contains 
.Prove that the circumcircles of triangles
and 
are coaxal with the circle with diameter 
.This post has been edited 1 time. Last edited by TheOverlord, May 11, 2015, 2:10 PM
is the incenter of 
then 
and 
Condition 
means that 
which is orthogonal to any circle through 
thus particularly orthogonal to the circumcircle 
of 
and similarly 
into 
takes the circle with diameter 
into 
(as 
If 
cut 
then from the complete quadrilateral 
it follows that 
But since 
bisect 
we have 
and 
and the circle with diameter 
be the Incenter and Circumcenter of 
.
and similarly 
.
of 
.
.
WRT 
.
.
where 
is the circumradius of 
.
around 
swaps 
(by Brocard), 
and 
remains fixed. So, 
.
and 
.
and 
and let 
concurs.
and 
respectively. So, 
.
is the Midpoint of 
.
must be the midpoint of Minor Arc 
.
are coaxial. 
