Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a convex quadrilateral with no pair of parallel sides, such that
![\angle ABC = \angle CDA](/media/m/d/7/b/d7bd5b7c92e42db2026c022013f3c73d.png)
. Assume that the intersections of the pairs of neighbouring angle bisectors of
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
form a convex quadrilateral
![EFGH](/media/m/5/4/6/546f6a8c4c38499f3e56b70541e9470d.png)
. Let
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
be the intersection of the diagonals of
![EFGH](/media/m/5/4/6/546f6a8c4c38499f3e56b70541e9470d.png)
. Prove that the lines
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
intersect on the circumcircle of the triangle
![BKD](/media/m/c/0/8/c085c6a13c99e3968015974c0b90c119.png)
.
%V0
Let $ABCD$ be a convex quadrilateral with no pair of parallel sides, such that $\angle ABC = \angle CDA$. Assume that the intersections of the pairs of neighbouring angle bisectors of $ABCD$ form a convex quadrilateral $EFGH$. Let $K$ be the intersection of the diagonals of $EFGH$. Prove that the lines $AB$ and $CD$ intersect on the circumcircle of the triangle $BKD$.