Giovanni Francesco Fagnano dei Toschi (born 31 January 1715 in

Senigallia Senigallia (or Sinigaglia in Old Italian, Romagnol: ''S’nigaja'') is a ''comune'' and port town on Italy's Adriatic coast. It is situated in the province of Ancona in the Marche region and lies approximately 30 kilometers north-west of the pro ...

, 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 Canon or Canons may refer to: Arts and entertainment * Canon (fiction), the conceptual material accepted as official in a fictional universe by its fan base * Literary canon, an accepted body of works considered as high culture ** Western ca ...

, and in 1755 he was appointed archdeacon of the cathedral of Senigallia.

Mathematics

Fagnano is known for

Fagnano's problem In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775: The solution is the orthic triangle, with vertices at the base points of the altitudes of the given triangle. Solution The orthic tri ...

, the problem of inscribing a minimum-

perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

within an

acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...

. 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 bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...

s are the altitudes of the original triangle. Fagnano also partially solved the problem of finding the

geometric median In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances ...

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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

, 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 In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that any set of ''d'' + 2 points in R''d'' can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these convex ...

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