000RM.dvi

11.6 Construction of a regular 17-gon 329

11.6 Construction of a regular 17-gon

To construct two vertices of the regular 17-gon inscribed in a given circle
O(A).

1.On the radiusOBperpendicular toOA, mark a pointJ such that

OJ=^14 OA.

2.Mark a pointEon the segmentOAsuch that∠OJE=^14 ∠OJA.

3.Mark a pointFon the diameter throughAsuch thatOis between

EandFand∠EJF=45◦.

4.WithAFas diameter, construct a circle intersecting the radiusOB

atK.

5.Mark the intersections of the circleE(K)with the diameter of

O(A)throughA. Label the one betweenOandApointsP 4 , and

the other andP 6.

6.Construct the perpendicular throughP 4 andP 6 to intersect the cir-

cleO(A)atA 4 andA 6.^5

ThenA 4 ,A 6 are two vertices of a regular 17-gon inscribed inO(A).
The polygon can be completed by successively laying off arcs equal to
A 4 A 6 , leading toA 8 ,A 10 , ...,A 16 ,A 1 =A,A 3 ,A 5 , ...,A 15 ,A 17 ,A 2.

(^5) Note thatP 4 is not the midpoint ofAF.