Construction of Regular Pentagon by H. W. Richmond
As Ptolemy's construction described by S. Brodie, the one below seeks to construct a regular pentagon inscribed in a given circle. The latter is dated 1893 and attributed to H. W. Richmond. The approach has been expanded by [Conway and Guy] to the construction of other regular polygons.
Let XA be a diameter of the circle with center O.
Choose P midway on the radius perpendicular to XA.
Draw a bisector of ∠APO to the intersection Q with XA,
and an external bisector to the intersection with XA at R.
If A is taken to be one of the vertices of the regular pentagon, Q and R are the projections of the other four onto XA. These can be obtained by erecting perpendiculars to XA at Q and R.
OQ / AQ = OP / AP = 1/2 : √5/2.
Since OQ + AQ = 1, we find
(This construction is easily Regular Pentagon Inscribed in Circle by Paper Foldingimplementable by paper folding.)
- J. H. Conway, R. Guy, The Book of Numbers, Springer, 1996
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