In the 1820s, Eugene Onegin is a bored Saint Petersburg dandy, whose life consists of balls, concerts, parties and nothing more. One day he inherits a landed estate from his uncle. When he moves to the country, he strikes up a friendship with his neighbor, a starry-eyed young poet named Vladimir Lensky. One day, Lensky takes Onegin to dine with the family of his fiancĂŠe, the sociable but rather thoughtless Olga Larina. At this meeting he also catches a glimpse of Olga's sister Tatyana. A quiet, precocious romantic and the exact opposite of Olga, Tatyana becomes intensely drawn to Onegin. Soon after, she bares her soul to Onegin in a letter professing her love. Contrary to her expectations, Onegin does not write back. When they meet in person, he rejects her advances politely but dismissively and condescendingly. This famous speech is often referred to as Onegin's Sermon: he admits that the letter was touching, but says that he would quickly grow bored with marriage and can only offer Tatyana friendship; he coldly advises more emotional control in the future, lest another man take advantage of her innocence.
Later, Lensky mischievously invites Onegin to Tatyana's name day celebration promising a small gathering with just Tatyana, Olga, and their parents. When Onegin arrives, he finds instead a boisterous country ball, a rural parody of and contrast to the society balls of St. Petersburg he has grown tired of. Onegin is irritated with the guests who gossip about him and Tatyana, and with Lensky for persuading him to come. He decides to avenge himself by dancing and flirting with Olga. Olga is insensitive to her fiancĂŠ and apparently attracted to Onegin. Earnest and inexperienced, Lensky is wounded to the core and challenges Onegin to fight a duel; Onegin reluctantly accepts, feeling compelled by social convention. During the duel, Onegin unwillingly kills Lensky. Afterwards, he quits his country estate, traveling abroad to deaden his feelings of remorse.
Tatyana visits Onegin's mansion, where she looks through his books and his notes in the margins, and begins to question whether Onegin's character is merely a collage of different literary heroes, and if there is, in fact, no "real Onegin".
Several years pass, and the scene shifts to St. Petersburg. Onegin has come to attend the most prominent balls and interact with the leaders of old Russian society. He sees the most beautiful woman, who captures the attention of all and is central to society's whirl, and he realizes that it is the same Tatyana whose love he had once turned away. Now she is married to an aged prince (a general). Upon seeing Tatyana again, he becomes obsessed with winning her affection, despite the fact that she is married. However, his attempts are rebuffed. He writes her several letters, but receives no reply. Eventually Onegin manages to see Tatyana and presents to her the opportunity to finally elope after they have become reacquainted. Does he desire her only for her wealth and position? She recalls the days when they might have been happy, but that time has passed. Onegin repeats his love for her. Faltering for a moment, she admits that she still loves him, but she will not allow him to ruin her and declares her determination to remain faithful to her husband. She leaves him regretting his bitter destiny.

Question: Why does Lensky challenge Onegin to a duel?
Answer: Lensky challenges Onegin to a duel, because Onegin flirts with Olga, who is actually attracted to Onegin.
Alternative answer: He is jealous of their flirting with eachother.