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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)
, let be the excircle opposite to 
. Let 
and 
be the points where is tangent to 
, and , respectively. The circle 
intersects line at 
and 
. Let 
be the midpoint of . Prove that the circle 
is tangent to .
be a rhombus. Let 
be a point on the line , other than 
or 
, such that 
. Let be the point of intersection of 
with the perpendicular bisector of . Prove that 
and are collinear.
be a rhombus. Let be a point on the line , other than or , such that . Let be the point of intersection of with the perpendicular bisector of . Prove that and are collinear.
met at . Since 
and 
and 
, 
is isosceles trapesoide, hence 
, therefor belongs perpendicular bisector of . is perpendicular bisector of 
, so is circumcenter of triangle , therefor belongs perpendicular bisector of 
, hence 
, so 
and are colinear.
