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Using a new point of view inspired by hyperplane arrangements, we generalizethe converse to Pascal's Theorem, sometimes called the Braikenridge-MaclaurinTheorem. In particular, we show that if 2k lines meet a given line, coloredgreen, in k triple points and if we color the remaining lines so that eachtriple point lies on a red and blue line then the points of intersection of thered and blue lines lying off the green line lie on a unique curve of degreek-1. We also use these ideas to extend a second generalization of theBraikenridge-Maclaurin Theorem, due to M\"obius. Finally we use Terracini'sLemma and secant varieties to show that this process constructs a dense set ofcurves in the space of plane curves of degree d, for degrees d <= 5. Theprocess cannot produce a dense set of curves in higher degrees. The expositionis embellished with several exercises designed to amuse the reader.

Generalizing the Converse to Pascal's Theorem via Hyperplane Arrangements and the Cayley-Bacharach Theorem