AoPS Academy
satisfies 
. If 
, find the maximum possible value of 
.
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
, two tangents are drawn touching at points 
and 
. A point 
is chosen on the segment 
. Let points 
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects again at point 
(
). Prove that the line 
passes through the centroid of triangle 
.
if known are the vertice 
the incenter 
and the intersection point of the medians 
if known are the vertice the incenter and the intersection point of the medians 
Let D be the (unknown) tangency point of the incircle (I) with BC. Then XI is the Newton line of degenerate tangential quadrilateral ABDC with the diagonals AD, BC and incircle (I), so that XI cuts AD at its midpoint Y. Hence one locus of D is a line 
passing through the reflection of A in XI. On the other hand, 
is right, so that another locus of D is a circle 
with diameter XI. This circle can intersect the line k in two points D, D', touch it in one point D, or miss the line completely. Since the midpoint X of BC and the tangency point D of the incircle (I) with BC must lie on the opposite sides of the bisector AI of 
, only the intersections of 
on the opposite side of AI than X are acceptable. Thus the problem can have two, one, or no solution. Once the point D is constructed, the line 
is known, the incircle (I) with radius r = ID is also known, and AB, AC are the incircle tangents from A intersecting XD at B, C.
and 
.
Where 
