the cayley-bacharach theorem says that if two plane cubics intersect in exactly 9 points, then any third cubic passing through eight of these must pass through the ninth. we'll give a weird, elementary but cute proof which shows something a tiny bit stronger. the prerequisites will be not nil but nilpotent, limited to bezout's theorem which i'll state carefully in the form i need. this proof came from math 262a, which apparently got it from terence tao's blog.