be an acute triangle with 
,
be the projection from 
.
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
sastifying 
Y by Adventure10
Three circles 
, 
and 
share a common point of intersection 
. The other common point of and is 
, that of and is 
, and that of and is 
. The line 
intersects the circle in the point 
. Similarly, the line 
intersects the circle in the point 
, and the line 
intersects the circle in the point 
. Show that
![\[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]](//latex.artofproblemsolving.com/d/d/3/dd3c9d7a2faf729c61dbeb213bcca2748c256248.png)
, 
and 
share a common point of intersection 
. The other common point of and is 
, that of and is 
, and that of and is 
. The line 
intersects the circle in the point 
. Similarly, the line 
intersects the circle in the point 
, and the line 
intersects the circle in the point 
. Show that![\[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]](http://latex.artofproblemsolving.com/d/d/3/dd3c9d7a2faf729c61dbeb213bcca2748c256248.png)
This post has been edited 1 time. Last edited by djb86, Apr 21, 2013, 2:39 PM
,
