be an acute triangle with 
,
be the projection from 
onto 
.
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](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 
.
.
sastifying 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
.
Y by Adventure10, Mango247
Let 
be a convex quadrilateral and let 
and 
be points in such that 
and 
are cyclic quadrilaterals. Suppose that there exists a point 
on the line segment 
such that 
and 
. Show that the quadrilateral is cyclic.
be a convex quadrilateral and let 
and 
be points in such that 
and 
are cyclic quadrilaterals. Suppose that there exists a point 
on the line segment 
such that 
and 
. Show that the quadrilateral is cyclic.This post has been edited 1 time. Last edited by parmenides51, Jul 13, 2018, 7:49 PM
Reason: added last sentence
Reason: added last sentence
