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is described on a circle 
, and is not a trapezius inscribed in a circle. Let the tangency points of and 
be 
respectively. A circle with a center 
, different from is tangent to the segement 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines . A circle with center 
, different from is tangent to segment 
and lines . Prove that the lines 
are concurrent.
define 
be the product of first 
primes. Determine if there exists positive integers 
with the following property :![\[ m?=n(n+1)(n+2)(n+3) \]](http://latex.artofproblemsolving.com/4/5/c/45cb4a00f9395b8c6edb08a343c3bbd664916a4d.png)
be reals such that 
and 
Prove that
Let be reals such that and 
Prove that
Let be reals such that and 
Prove that
be on the minor arc . Let 
be tangents to the incircle, with 
on the circumcircle and 
is on the arc 
and 
is on the arc . Then it suffices to show 
is tangent to the incircle. By Pascal's on hexagon 
, we get the intersection of 
, call it 
, the intersection of 
, call it 
and the intersection of 
, call it 
, are collinear. In short, 
are collinear. Now use the converse of Brianchon on 
, since 
are concurrent, and 
are tangent to the incircle, we have is tangent to the incircle, and we are done.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
, 
be the circumcircle and inscribed circles of triangl
,we can construct a triangle 
such that 
