MathDB

An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon

P_1P_2P_3P_4P_5

in a circle. Now choose an arror bound

\epsilon > 0

and apply the following procedure. (a) Denote

P_0 = P_5

and

P_6 = P_1

and construct the midpoint

Q_i

of the circular arc

P_{i-1}P_{i+1}

containing

P_i

. (b) Rename the vertices

Q_1,...,Q_5

P_1,...,P_5

. (c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs

P_iP_{i+1}

is less than

\epsilon

. Prove this procedure must end in a finite time for any choice of

\epsilon

and the points

P_i

Problem Statement