There is no contradiction between Imam Ali (as) not defending Lady Fatimah (as) and her being offended and oppressed and between his courage and strength and being there when it all took place. The reason is that in such a situation, Imam Ali (as) had only two choices; he could either unsheathe his sword and begin to fight in defense of his and his families rights (which one of those rights was to be true successor to the Prophet (pbuh) after him), or to stay silent and begin to strengthen Islam in other fields (even though he had been deprived of being khalifah after the Prophet (pbuh)). Since the first option would result in the disunion and division of the Muslim nation, thus making it prone to attacks by the Roman Empire or those opposing Islam (e.g. the apostates and those who falsely claimed they were divine prophets after Prophet Muhammad (pbuh)), he had no option but to go with the second and sacrifice his wife and child’s lives in order for the Prophet’s efforts and the blood of the martyrs not to be wasted and for Islam to live on.
When a group of the Prophet’s (pbuh) close companions had gathered in Imam Ali’s (as) home and wouldn’t pledge allegiance (bay’ah) to Abu Bakr in objection to the decisions that had been made in Thaqifah (the most important one being the caliphate of Abu Bakr), Umar was chosen on Abu Bakr’s behalf to get the bay’ah from them and especially from Ali (as). It is well documented that in the process of carrying out Abu Bakr and Umar’s plans, some incidents that shouldn’t have taken place took place that you can learn about by referring to reliable sources.
Unfortunately, according to various dependable narrations that have been documented in history that have even been recorded in Sunni sources, this tragedy has taken place (although there be slight differences concerning its details).
As for why Imam Ali (as) stayed silent regarding what was taking place, despite his unique and unparallel courage and strength, one must say:
Imam Ali (as) could only make one of two choices; one was to rise against the new government with the help of his true friends and companions who were against it, and make it the way it was supposed to be, or tolerate the situation that had come up and to solve the problems of the Muslims and carry out his personal responsibilities as much as possible without causing any conflict.
Considering the fact that divine leaders don’t consider power and ranks their main interest and goal and have much higher goals, after contemplating the political and social conditions of the Muslim nation at the time, Imam Ali (as) reached the conclusion that if he were to insist on his rights regarding the oppression that had been done towards Lady Fatimah (as), it would end in the dissipation of the great efforts of the Prophet (pbuh) and the blood of the martyrs that had been shed in his way. Therefore, he decided to accept what had taken place and stay quiet and to protect and take care of Islam indirectly.
If his excellency had chosen to fight instead, or in other words, if he had chosen to exhibit his bravery and strength, the government surely would have reacted correspondingly, resulting in a civil war among the Muslims and many bitter incidents that would take place, such as:
1- Imam Ali (as) would lose many of his true companions who had chosen to fulfill their pledge to the Prophet (pbuh) and stay with him after his demise who deeply loved Ali (as).
2- A great disunion between the Muslims would take place and a great number of sahabis (companions of the Prophet (pbuh)), who were considered a power in the face of the enemies of Islam, would be lost, or at the least, the Muslim nation would be weakened by such a disunion.
3- A great number of those who had converted to Islam during the final years of the Prophet’s (pbuh) life would turn away from Islam and rise against the Muslim government, endangering the Muslim nation.
4- There were chances that those falsely claiming prophethood after Prophet Muhammad (pbuh), such as Musaylamah and others, would try to harm Islam in different ways.
What was said was a few of the many reasons for why Imam Ali (as) preferred to be patient and silent instead of fighting. It was through his decision that the Muslim nation remained strong and didn’t become prone to the many dangers that were surrounding it. His decision was one that protected Islam itself and helped maintain the unity of the Muslims and was a fulfillment of his duty (meaning that this decision was his duty and that it was what Allah (swt) and the Prophet (pbuh) had wanted from him and he wasn’t allowed to make any other decision that would harm Islam and the Muslim nation or put them to danger).
Imam Ali (as) made this point to Lady Fatimah (as) when she had said to him that: “Oh son of Abu Talib!...You are the one who would break the wings of hawks (signifying his strength), but now you aren’t even capable of breaking the wings of chickens!” while delivering a sermon in the Mosque of the Prophet (pbuh).
With a tone full of sadness and pain, Imam Ali (as) unveils the secret behind his silence in the sermon of Shiqshiqiyyah of the Nahjul-Balaghah saying:
“Then I began to think whether I should assault or endure calmly the blinding darkness of tribulations wherein the grown up are made feeble and the young grow old and the true believer acts under strain till he meets Allah (on his death). I found that endurance thereon was wiser. So I adopted patience although there was pricking in the eye and suffocation (of mortification) in the throat. I watched the plundering of my inheritance…”
On another occasion, when one of his followers was counting Imam Ali’s (as) virtues for those present in Thaqifah (the place where the decisions for the caliphate of Abu Bakr were made), and was scolding them for putting Imam Ali (as) aside, his excellency said: “The safety of Islam (from dispute and disunion) is more important to us than anything else.”
These statements show that the thing that had pushed Imam Ali (as) to stay quiet and not object was of more importance than the bitter things that took place for him and his family, causing him to sacrifice them for its well-being, which was the safety of Islam.
 History of Tabari, vol.3, pg.202: “اتی عمر بن خطاب منزل علی(ع) فقال : لاحرقن علیکم او لتخرجن الی البیعة” Ibn abil-Hadid has also quoted this sentence in his commentary of the Nahjul-Balaghah (vol.2, pg.56) from the book Thaqifah of Jawhari; Al-Imamah wal-Siyasah, vol.2, pg.12; Ibn abil-Hadid, Sharh Nahjil-Balaghah, vol..1, pg.134; A’lamul-Nisa, vol.3, pg.1205. Ibn Abd Rabbih Andulusi (495 ah) says:
بعث الیهم ابوبکر عمر بن خطاب لیخرجهم من بیت فاطمة و قال له ان ابوا فقاتلهم. فاقبل بقبس من النار علی ان یضرم علیهم الدار. فلقیته فاطمة فقالتیابن الخطاب اجئت لتحرق دارنا؟ قال:نعم او تدخلوا فیما دخلت فیه الامة
Aqdul-Farid, vol.4, pg.260. Also see: History of Abil-Fada’, vol.1, pg.156; A’lamul-Nisa, vol.3, pg. 1207. Ithbatul-Wasiyyah, al-Mas’udi 124:
علیه و احرقوا بابه و استخرجه منه کرها و ضغطوا سیدة النساء بالباب حتی اسقطت محسنا (Mihantu-Fatimah, pg.60)
Milal wa Nihal, vol.2, pg.95; Talkhisul-Shafi, vol.3, pg.76, Furghe Velayat, pg.186.
 For further information, see: 1- The book “Pasokh be Afsaneye Shahadat” of Seyyid Jawad Hoseini Tabatabai 2- The Port of Answering Religious Questions website 3- The Balagh website.
 Mahdi Pishva’I, Sireye Pishvayan, pg.65.
 Ibid, pg. 71.
 Izzuddin Hoseini, Sharhe Khutbeye Hazrate Zahra, vol. 2, pg332.
 Nahjul-Balaghah, sermon 3.
 Nahjul-Balaghah of Subhi Saleh, sermon 3.
 See: Sharhe Nahjul-Balaghah, vol.6, pp. 23-45.