be a scalene acute triangle with incenter 
and circumcircle 
.
is the midpoint of small arc 
o
is the projection of 
and 
.
with center 
is internally tangent to 
,
meets 
,
bisects 
.
be an acute-angled scalene triangle with circumcenter 
.
,
,
the midpoints of sides 
,
,
,
be the circle passing through 
and tangent to 
at 
at points 
and 
,
be the midpoint of segment 
,
be the intersection of lines 
.
and that triangle 
is isosceles.
to 
such that 
is an integer?
Y by Adventure10, Mango247, and 2 other users
In the following, the point of intersection of two lines 
and 
will be abbreviated as 
.
Suppose
is a triangle in which 
and 
. Let 
be the circumcircle of the triangle . Let 
and 
be the tangents to the circle at 
and 
, respectively.
Let
and 
. Furthermore, let 
, and let 
. Denote by 
the foot of the perpendicular from 
on . Denote by 
the point of intersection of the line 
with the circle (different from 
). Denote by 
be the point of intersection of the line 
with the circle (different from ). Finally, define 
. Prove that
![\[ \frac {SU \cdot SP}{TU \cdot TP} = \frac {SA^{2}}{TA^{2}}.
\]](//latex.artofproblemsolving.com/8/0/c/80ce6726a475d2b98b418f9d72e4d3df89d67b5a.png)
and 
will be abbreviated as 
.Suppose
is a triangle in which 
and 
. Let 
be the circumcircle of the triangle . Let 
and 
be the tangents to the circle at 
and 
, respectively.Let
and 
. Furthermore, let 
, and let 
. Denote by 
the foot of the perpendicular from 
on . Denote by 
the point of intersection of the line 
with the circle (different from 
). Denote by 
be the point of intersection of the line 
with the circle (different from ). Finally, define 
. Prove that![\[ \frac {SU \cdot SP}{TU \cdot TP} = \frac {SA^{2}}{TA^{2}}.
\]](http://latex.artofproblemsolving.com/8/0/c/80ce6726a475d2b98b418f9d72e4d3df89d67b5a.png)
neet 
at point 
,by Pascal Theory we get that 
,
is tagent to 
,
;
;
;
;
;
QED. 
= 
= 
.
= 
.
