Show that a point lies on the diagonal of quadrilateral

Question

In a quadrilateral ABCD we choose a point E on the side AD and a point F on the side CD. Then we choose a point G on the line EF. Let H be the second point of the intersection between the circles that have the triangles EGD and BGF inscribed. Show that the point H lies on the diagonal BD.

Can anyone solve this?

Answer 1

Consider the problems the other way round. Start by drawing two intersecting circles and label the points of intersection $G$ and $H$.

Now draw a straight line through $G$ meeting one circle at $E$ and the other at $F$.

Now choose points $D$ on the $E$ circle and $B$ on the $F$ circle.

Clearly we can construct this so that $DHB$ is not a straight line, and obviously we can construct the points $A$ and $C$, thus fulfilling the rubric of the question.

Therefore the hypothesis as originally stated is not generally true.