Abstract / Synopsis
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 proof gets to the heart of the matter more efficiently, the synthetic proof does a better job of revealing hidden aspects of the geometric configuration. Moreover, as Pappus presents it, the synthetic proof provides a real element of surprise and a sense of discovering unexpected connections. We conclude with some general observations about synthetic versus algebraic techniques in geometry and in the teaching and learning of mathematics.
John B. Little, "One Theorem, Two Ways: A Case Study in Geometric Techniques," Journal of Humanistic Mathematics, Volume 13 Issue 2 (July 2023), pages 338-350. DOI: 10.5642/jhummath.CQBL1536. Available at: https://scholarship.claremont.edu/jhm/vol13/iss2/18