What are a wife's rights after her husband has abandoned her for three years, married another woman without her knowledge, and had children with her? Is she considered divorced or not? Does she have the right to demand a divorce if he returns? And how should the shared home, to which she contributed the larger portion of its price, be divided?

The mother is the father's wife unless he divorces her, and she has full rights, such as maintenance, clothing, and companionship. If the father defaults on these, the mother can demand her rights, or be patient, or seek a divorce. If he refuses, she can seek khulʿ (divorce at the wife's instance with compensation), noting that seeking divorce without a valid reason is forbidden. Your primary duty is to strive for reconciliation between your parents, and to remind your father that even if he marries, he is not entitled to oppress your mother or deny her rights, especially good companionship; otherwise, he will bear the burden of that before Allah. A husband must be just between his wives in terms of maintenance, clothing, and equal division of time (qasm), as the Prophet, peace and blessings be upon him, used to do. Whoever inclines to one of them will come on the Day of Resurrection with one side of his body drooping. As for love and sexual intercourse, there is no blame on him concerning them, due to Allah's saying: ﴿And you will never be able to do justice between wives, even if you should strive to do so.﴾ (An-Nisa: 129). If the husband wishes to divorce one of them, he may do so. If they agree that the wife remains without an equal division of time (qasm) by her consent, it is permissible, due to Allah's saying: ﴿And if a woman fears from her husband desertion or neglect, there is no blame upon them if they make terms of settlement between them - and settlement is best.﴾ (An-Nisa: 128).