AoPS Academy
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
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.
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 a cyclic quadrilateral. Let 
be a point on the ray 
which is outside the circumscribed circumference of . Suppose that 
. Prove that 
if and only if 
.
be a cyclic quadrilateral. Let be a point on the ray which is outside the circumscribed circumference of . Suppose that . Prove that if and only if .
?
so we conclude that 
if 
we get that 
so 
then 
is isosceles and one more time 
)
is isosceles triangle with 
.
and 
we get 
,