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
Giulio Carlo, Count Fagnano, Marquis de Toschi (26 September 1682 — 18 May 1766) was an Italian mathematician. He was probably the first to direct attention to the theory of elliptic integrals. Fagnano was born in Senigallia (at the time spelled ...
, also a mathematician.
Religious career
Fagnano was ordained as a priest. In 1752 he became
canon,
and in 1755
he was appointed
archdeacon
An archdeacon is a senior clergy position in the Church of the East, Chaldean Catholic Church, Syriac Orthodox Church, Anglican Communion, St Thomas Christians, Eastern Orthodox churches and some other Christian denominations, above that o ...
of the cathedral of Senigallia.
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. Another property of the orthic triangle, also proven by Fagnano, is that its
angle bisectors are the altitudes of the original triangle.
Fagnano also partially solved the problem of finding the
geometric median of sets of four points in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
; 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.
[http://www.izwtalt.uni-wuppertal.de/Acta/NAE1775.pdf ]
References
{{DEFAULTSORT:Fagnano, Giovanni
1715 births
1797 deaths
Italian mathematicians
18th-century Italian mathematicians
18th-century Italian Roman Catholic priests
People from Senigallia