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.

Proof

Assume the radius of the circle is 1. Then OP = 1/2. From the Pythagorean theorem, AP = √5/2. By a property of angle bisectors,

OQ / AQ = OP / AP = 1/2 : √5/2.

Since OQ + AQ = 1, we find OQ = (√5 - 1) / 4. But the latter is the value of cos(72°), which allows us to conclude (from ΔOBQ) that ∠BOQ = 72°, such that A and B are indeed successive vertices of a regular pentagon.

Similarly, but using a property of external angle bisectors, OR = (√5 + 1) / 4, which is cos(36°). So that ∠COR = 36° and C is the next (after B) vertex of the regular pentagon.

(This construction is easily Regular Pentagon Inscribed in Circle by Paper Foldingimplementable by paper folding.)

References

J. H. Conway, R. Guy, The Book of Numbers, Springer, 1996

Approximate Construction of Regular Pentagon by A. Durer
Construction of Regular Pentagon by H. W. Richmond
Inscribing a regular pentagon in a circle - and proving it
Regular Pentagon Construction by Y. Hirano
Regular Pentagon Inscribed in Circle by Paper
Mascheroni Construction of a Regular Pentagon
Regular Pentagon Construction by K. Knop

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