satisfying
Prove
,
,
be four points on a line, in that order. Let 
be one of the intersections of the circle with diameter 
,
.
,
and 
are three points, I will call their circumcircle 
.
and 
are tangent, that 
and 
are tangent, and that the first two circumcircles are orthogonal to the last two circumcircles.
is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
Y by Adventure10
Let 
be a scalene (i.e. non-isosceles) triangle. Let 
be the center of the circumcircle of this triangle and 
the center of the incircle. Assume that the second point of intersection different from 
of the angle bisector of 
with the circumcircle of lies on the perpendicular bisector of 
.
Show that
is the second-largest angle in the triangle .
be a scalene (i.e. non-isosceles) triangle. Let 
be the center of the circumcircle of this triangle and 
the center of the incircle. Assume that the second point of intersection different from 
of the angle bisector of 
with the circumcircle of lies on the perpendicular bisector of 
.Show that
is the second-largest angle in the triangle .This post has been edited 1 time. Last edited by Martin N., May 24, 2012, 7:11 AM
not containing 
is the power of the incenter with respect to the circumcircle.
) is a 30-60-90 triangle in that order, <ACI=30 then. Similarly, <ICB=30, so <ACB=60, and gamma is obviously the middle angle.
and the condition in the problem statement gives 
.
is cyclic and hence 
,
,
