Suppose f is the cubic polynomial vanishing on the three lines through AB, CD, EF and g is the cubic vanishing on the other three lines BC, DE, FA . Pick a generic point P on the conic and choose λ so that the cubic h = f + λg vanishes on P . Then h = 0 is a cubic that has 7 points A, B, C, D, E, F, P in common with the conic. But by Bézout's theorem a cubic and a conic have at most 3 × 2 = 6 points in common, unless they have a common component. So the cubic h = 0 has a component in common with the conic which must be the conic itself, so h = 0 is the union of the conic and a line. It is now easy to check that this line is the Pascal line.
Scholarly interest in this matter involves more than just idle curiosity and medical detective-work. The question of Pascal’s physical and mental health goes to the heart of desires to learn more about the conditions and circumstances that produce extraordinary genius. Affliction and disease, physical or emotional trauma, a natural disadvantage or disability have often served as an added motive or accelerator for high-level creative achievement. Examples abound – from the ancient legend of the blind and vagabond Homer to the documented histories of modern creative figures like Isaac Newton, Van Gogh, Stephen Hawking, and Christy Brown. We can only speculate whether and to what extent Pascal’s physical ailments and disabilities, instead of retarding his career, may have actually spurred and given rise to his intellectual triumphs.
This is the same advice Dostoevsky's guru, Father Zossima, gives to the "woman of little faith" in The Brothers Karamazov . The behavior Pascal mentions is "taking holy water, having Masses said, and so on". The behavior Father Zossima counsels to the same end is "active and indefatigable love of your neighbor." In both cases, living the Faith can be a way of getting the Faith. As Pascal says: "That will make you believe quite naturally and will make you more docile." "But that is what I am afraid of.'' ''But why? What have you to lose?"