# 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

Similarly, but using a property of external angle bisectors,

(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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