> 
that satisfy 
,
€ 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
such that 
.
and 
.
Y by Adventure10, Mango247
Points 
and 
are isogonal conjugates in the traingle 
. We draw perpendiculars 
, 
, 
, and 
, 
, 
to the sides 
, 
, 
respectively. Let 
, 
, and 
, 
and 
, 
intersect at 
, 
, 
respectively. Show that the lines 
, 
, and 
are parallel.
and 
are isogonal conjugates in the traingle 
. We draw perpendiculars 
, 
, 
, and 
, 
, 
to the sides 
, 
, 
respectively. Let 
, 
, and 
, 
and 
, 
intersect at 
, 
, 
respectively. Show that the lines 
, 
, and 
are parallel.
which will obviously solve the problem. Take 
to be the projection of 
on 
.
lie on a circle centered in 
,
(where 
stands for the circle with diameter 
and 
stands for the circumcircle of 
)
which are concurrent, so 
passes through 
so 
is cyclic.The radical axis of 
and 
and 
is a point on the radical axis of 
and 
.
is the radical axis of the two circles so the centre joining line is perpendicular to 
.
and 
so the conclusion follows.
