The question is from Kiselev's Geometry Book 1, exercise 220. I am not even sure from where I should start the construction.
When analyzing the problem, I tried to use the fact that the diagonal BD is divided by the diagonal AC in the ratio of the segments AB and AD since AC is the bisector, but I doubt it can be used to facilitate the construction since the diagonals are not given segments. The chapter introduces the method of reflection and translation, which I could not apply to the problem. Any help would be appreciated.
Recall that the chapter is on reflection, and we have angle bisectors, so is there a natural reflection to use?
Hint: WLOG $AB > AD$. Find a point $D^*$ on $AB$ and a point $C$ such that $AD = AD^*$ and $ CD = CD^*$.