Regular polygons

In this session , the problems consists the uses of symmectries of the polygon . Here , we illustrate this with the following fasciating facts of the construction of the regular petagons .Of cource there exists a classical ruler and compass construction , but there is an easierway to do it . At first , make the simplest knot, the trefoil knot, on a ribbon of paper , then flattern it as the following knot .After cutting of the two ends of the ribbon , one obtains a regular pentagons .
Problems :
1.Let $A_1A_2....................A_n$ be a regular polygons inscribed in the circle of the centre $O$ and the radius $R$. On the half line $OA_1$ cnoose $P$ such that $A_1$ is between $O$ and $P$ .Prove that
$\displaystyle{\prod_{i=1}^{n}}PA_i$ = $PO^n-R^n $.
2.Let $A_1...........A_7$ be a regular heptagon .Prove that
$\frac{1}{A_1A_2} = \frac{1}{A_1A_3} + \frac{1}{A_1A_4}.$
If the above relation holds then prove that the polygon is a heptagon , that is $ n = 7$( $INMO-1992$).

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