Abstract
An almost forgotten gem of Gauss tells us how to compute the area of a
pentagon by just going around it and measuring areas of each vertex triangles
(i.e. triangles whose vertices are three consecutive vertices of the pentagon).
We give several proofs and extensions of this beautiful formula to hexagon etc.
and consider special cases of affine--regular polygons.
The Gauss pentagon formula is, in fact, equivalent to the Monge formula which
is equivalent to the Ptolemy formula.
On the other hand, we give a new proof of the Robbins formula for the area of
a cyclic pentagon in terms of the side lengths, and this is a consequence of
the Ptolemy formula. The main tool is simple: just eliminate from algebraic
equations, via resultants. By combining Gauss and Robbins formulas we get an
explicit rational expression for the area of any cyclic pentagon. So, after
centuries of geometry of triangles and quadrilaterals, we arrive to the
nontrivial geometry of pentagons.
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