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Full-Text Articles in Physical Sciences and Mathematics
One Theorem, Two Ways: A Case Study In Geometric Techniques, John B. Little
One Theorem, Two Ways: A Case Study In Geometric Techniques, John B. Little
Journal of Humanistic Mathematics
If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, and K on ΓA cutting those sides in same ratios:
AH : HB = BΘ : ΘΓ = ΓK : KA,
then Pappus of Alexandria proved that the triangles ABΓ and HΘK have the same centroid (center of mass). We present two proofs of this result: an English translation of Pappus's original synthetic proof and a modern algebraic proof making use of Cartesian coordinates and vector concepts. Comparing the two methods, we can see that while the algebraic …