Let 
be a any quadrilateral. 
are intersection of angle bisector of four vertex 
(Show in the figure attachment. M,N,P,Q are intersection of four line through and pependicular with 
(show in the figure).
http://www.geogebratube.org/student/m79112
1-Prove that
are tangential quadrilateral.
2-Prove that if is cyclic, then is a point 
, and center of circle 
, and intersection of 
are collinear
Solution by: Luis González
be a any quadrilateral. are intersection of angle bisector of four vertex (Show in the figure attachment. M,N,P,Q are intersection of four line through and pependicular with (show in the figure).http://www.geogebratube.org/student/m79112
1-Prove that
are tangential quadrilateral.2-Prove that if
is cyclic, then is a point , and center of circle , and intersection of are collinearSolution by: Luis González
1) Label
the angles of 
Then 
and 
Similarly we have 
is cyclic and let be its circumcenter.Since
we get 
is P-isosceles 
is perpendicular bisector of 
bisects 
Similarly, 
bisect 
is tangential with incenter 
2) Let
and WLOG assume that 
is incenter of 
and 
is X-excenter of 
Then 
When is cyclic, we have then 
is cyclic, i.e. 
is antiparallel to 
WRT 
perpendicular from 
to goes through the circumcenter of 
Similarly, perpendiculars from 
to 
pass through 
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