Giovanni Francesco Fagnano dei Toschi (born 31 January 1715 in Senigallia, died 14 May 1797 in Senigallia) was an Italian churchman and mathematician, the son of Giulio Carlo de' Toschi di Fagnano, also a mathematician.[1]
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Fagnano's Problem by Unfolding
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Fagnano's Problem Analytical Solution
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Religious career
Fagnano was ordained as a priest. In 1752 he became canon,[1] and in 1755[1] he was appointed archdeacon of the cathedral of Senigallia.[2]
Mathematics
Fagnano is known for Fagnano's problem, the problem of inscribing a minimum-perimeter triangle within an acute triangle. As Fagnano showed, the solution is the orthic triangle, whose vertices are the points where the altitudes of the original triangle cross its sides.[3] Another property of the orthic triangle, also proven by Fagnano, is that its angle bisectors are the altitudes of the original triangle.[1]
Fagnano also partially solved the problem of finding the geometric median of sets of four points in the Euclidean plane; this is the point minimizing the sum of its distances to the four given points. As Fagnano showed, when the four points form the vertices of a convex quadrilateral, the geometric median is the point where the two diagonals of the quadrilateral cross each other. In the other possible case, not considered by Fagnano, one point lies within the triangle formed by the other three, and this inner point is the geometric median. Thus, in both cases, the geometric median coincides with the Radon point of the four given points.[4][5][6][7]
References
- ^ a b c d O'Connor, John J.; Robertson, Edmund F., "Giovanni Francesco Fagnano dei Toschi", MacTutor History of Mathematics Archive, University of St Andrews
- ^
This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). "Giulio Carlo de' Toschi di Fagnano". Catholic Encyclopedia. New York: Robert Appleton Company.
- ^ Gutkin, Eugene (1997), "Two applications of calculus to triangular billiards", The American Mathematical Monthly, 104 (7): 618–622, doi:10.2307/2975055, JSTOR 2975055, MR 1468291.
- ^ Cieslik, Dietmar (2006), Shortest Connectivity: An Introduction with Applications in Phylogeny, Combinatorial Optimization, vol. 17, Springer, p. 6, ISBN 9780387235394
- ^ Plastria, Frank (2006), "Four-point Fermat location problems revisited. New proofs and extensions of old results" (PDF), IMA Journal of Management Mathematics, 17 (4): 387–396, doi:10.1093/imaman/dpl007, Zbl 1126.90046
- ^ Fagnano, G. F. (1775), "Problemata quaedam ad methodum maximorum et minimorum spectantia", Nova Acta Eruditorum: 281–303.
- ^ http://www.izwtalt.uni-wuppertal.de/Acta/NAE1775.pdf [bare URL PDF]
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This page was last edited on 18 February 2024, at 12:50