[/quote]
,
and 
.
Where 
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
,
and 
.
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
be an acute-angled triangle with 
and let 
-
.
and 
respectively. A line through 
and 
respectively such that 
and 
and 
are tangent to each other.
with circumcircle 
.
is the angle bisector of 
.
intersects 
.
angle bisector of 
if and only if 
.
be real numbers satisfying
.
,
past 
.
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
and 
intersect 
and 
at 
intersect 
.
with orthocenter 
intersects 
at 
intersect 
.
intersects 
and 
intersect on the circumcircle of triangle 
.
.![$D \in [AB]$](http://latex.artofproblemsolving.com/0/7/9/07914568284f3822cf15730a6a0e868f0bd79996.png)
is a point. Perpendiculars from ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
and ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
intersects with ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
intersects with 
and 
.![$$[E_1F_1]=\frac{3}{4}$$](http://latex.artofproblemsolving.com/3/e/f/3efdf53cb3ff4b2a7fa60691f5aac9b3eeecf00c.png)
and 
.
be the point on 
.
Y by aayush-srivastava, Adventure10, Mango247
Given four points 
on a circle such that 
is a diameter and 
is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points 
and with the point of intersection of the lines 
and 
is perpendicular to the line 
on a circle such that 
is a diameter and 
is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points 
and with the point of intersection of the lines 
and 
is perpendicular to the line 
.
,
,
on a circle, the point of intersection of the lines 
and 
lies on the polar of the point of intersection of the lines 
and 
with respect to the circle. Applying this to our points A, C, D, B which lie on our circle, we see that the point of intersection of the lines AC and DB lies on the polar of the point of intersection of the lines CD and BA with respect to our circle. In other words, the point F lies on the polar of the point P with respect to our circle. The point E also lies on the polar of the point P, since the point E is actually (par definitionem) the pole of the line CD, and as the line CD passes through the point P, the pole E of the line CD lies on the polar of the point P.
,
.
,
and Let 
,
,
,
,
,
,
is the polar of 
,
,
is also the polar of 
Be the intersection of 
and 
.
.
are collinear and 
.
are concyclic. From 
,
.
- orhocenter of ![$[AC\cap BD]AB$](http://latex.artofproblemsolving.com/0/a/c/0acb0c154597098c551f549eeafc8eb895be2e43.png)
)
, 
are collinear in the polar of 
.
.
