Convex quadrilaterals and Tangents

Question

Given a convex quadrilateral $ABCD$, we neeed to prove that the circles inscribed in the triangles $ABC$ and $CDA$ are tangent if and only if the circles inscribed in the triangles $BCD$ and $DAB$ are tangent.

Any hints

Thanks

Answer 1

Hint: If the incircles of $ABC$ and $CDA$ are tangent at a common point on the diagonal $AC$ then $ABCD$ has an incircle. Then look at the diagonal $BD$ and denote the points of tangency of the incircles $ABD$ and $BCD$ with the diagonal $BD$ by $Q_1$ and $Q_2$. Then show that $BQ_1 = BQ_2$ which is equivalent to $Q_1 \equiv Q_2 \equiv Q$.