I've been trying to solve this problem proposed as part of one of the first lectures of a Berkeley linear algebra course:
"What Good is a Basis ? The freedom to choose a basis often simplifies calculations and proofs. For instance, here is a phenomenon first noticed by G. Desargues (1593 - 1662), a contemporary of R. Descartes: In the plane, fix two intersecting straight lines B and C and a point p on neither. Through p draw two straight lines X and Y that intersect B and C in four points all told. Two pairs of those points are not yet joined by straight lines; draw those lines now and, if they intersect, call their intersection q . As X and Y move, always passing through p , so does q move; show that it moves along some fixed straight line D . To prove the existence of D takes some ingenuity if none but the methods of Euclidean plane geometry may be used; and if rectangular Cartesian coordinates must be used the proof is a tedious computation. But a relatively short computation suffices if we choose an apt basis. Test these claims by trying to verify Desargues’ observation above using only the ideas you learned in High-School. Then you will be better able to appreciate the strategy motivating vector notation and its algebra used in the following proof."
The construction suggested reminds me of concepts like Ceva's theorem, complete quadrangle, internal/external division, which I learned about in Geometria Moderna (Modern Geometry required subject for math majors at UNAM Mexico). Sadly, though, I can't think of a way to use any of those concepts to solve the problem. Suggestions, please!