id: 05694772
dt: j
an: 2010b.00434
au: Percy, Andrew; Rogers, D.G.
ti: Alternative route: from van Schooten to Ptolemy.
so: Normat. 57, No. 3, 116-128 (2009).
py: 2009
pu: Nationellt Centrum för Matematikutbildning (NCM), Göteborgs Universitet,
Göteborg
la: EN
cc: G40 A30
ut: Ptolemy’s theorem; distance problems; historical survey; history of
mathematics
ci:
li:
ab: Summary: A cyclic quadrilateral is a polygon with four vertices, all of
which lie on a circle. Such enjoy some special properties. It is
well-known that the sum of opposite angles always add up to $π$, it is
maybe less well-known that the rectangle formed by the diagonals has
the same area as the sum of the rectangles made up by opposite sides.
The latter is known as Ptolemy’s theorem. It has many consequences,
not only of trigonometric computations, but also of justifying the
elegant solution of the Dutch mathematician van Schooten (1615‒60) of
constructing the minimal sum of distances from a point to the vertices
of a triangle, a problem posed as a challenge by Fermat. The article
gives a historical survey and indicates how van Schooten’s
construction could serve as an inspiration by ‘cutting and pasting’
to suggest and prove Ptolemy’s theorem.
rv: