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22. If from any point within a regular polygon of n sides perpendiculars are drawn to all of the sides, the sum of these perpendiculars is equal to n times the apothem of the polygon.

SUGGESTION. - Join the point to each vertex. Then get two expressions for the area of the polygon, and put them equal to each other.

23. The stock from which tools are made is often in the form of round or cylindrical rods. From a round rod a tool is to be cut so that one end is a regular hexagon whose side is in. What size (diameter) of stock must be selected from which to cut it?

256. Construction. - Inscribe a square in a given circle.

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Given a circle with center O.

Required to inscribe a square in the circle.

Construction. 1. Draw two perpendicular diameters, AC and BD. Draw AB, BC, CD, and DA.

2. Then ABCD is the required square. Proof. The proof is left to the student.

257. Corollary 1. - Regular polygons of 4, 8, 16, 32, etc., sides may be inscribed in a circle.

How? Use § 247.

258. Corollary 2. By drawing tangents at the vertices of regular inscribed polygons of 4, 8, 16, 32, etc., sides, regular polygons of the same numbers of sides may be circumscribed about a circle.

This follows from § 250 and § 257.

259. Construction. - Inscribe a regular hexagon in a given circle.

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Given a circle with center O.

Required to inscribe a regular hexagon in the circle.

Construction. 1. Mark any point A on the circle. With center at A and radius equal to that of the circle, draw an arc intersecting the circle at B.

2. Draw AB.

3. Then AB is a side of the required hexagon, and by applying it six times as a chord, the regular hexagon ABCDEF may be inscribed.

Suggestions. AOAB is equilateral. Hence ∠AOB=60°, and is contained exactly six times about O. Why? Prove that AB is contained exactly six times in the circle. Apply § 245.

Make the construction, and write out the construction and proof.

260. Corollary 1. - By joining the alternate vertices of a regular inscribed hexagon, an equilateral triangle may be inscribed in a circle.

The proof is left to the student. Make the construction.

261. Corollary 2. - Regular polygons of 3, 6, 12, 24, etc., sides may be inscribed in a circle.

How? Use § 247.

262. Corollary 3. - By drawing tangents at the vertices of regular inscribed polygons of 3, 6, 12, 24, etc., sides, regular polygons of the same numbers of sides may be circumscribed about a circle.

This follows from § 250 and § 261. Explain.

263. Construction. - In cribe a regular decagon in a given circle.

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Given a circle with center 0.

Required to inscribe a regular decagon in the circle.

Construction. 1. Draw any radius OA, and divide it in extreme and mean ratio at M.

2. Beginning at A, mark off ten arcs AB, BC, etc., with chords each equal to OM, the longer segment of OA. Draw the chords of these arcs.

3. These chords form the required regular decagon ABCD....

Proof. 1. Draw OB and MB.

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12. ZOBA = 2 x and BAO = 2 ∠ x.

2x.

Ax. XII

13. ... in A ABO, 5 x = 180°.

§ 48

14. ... ∠ x = 36°.

Ax. V

15... AB = 36°, or exactly one tenth of the circle. §173

16. Since chords AB, BC, CD, etc., were constructed

equal, AB = BC = CD = etc.

§ 151, (7)

17. ... each arc, BC, CD, etc., equals one tenth of the

Ax. I

§ 245

circle, or the ten arcs constitute the circle.

18. ... ABCD is a regular decagon.

...

264. Corollary 1. - By joining the alternate vertices of a regular inscribed decagon, a regular pentagon may be inscribed in a circle.

Make the construction and give proof.

265. Corollary 2. - Regular polygons of 5, 10, 20, 40, etc., sides may be inscribed in a circle.

How? Give proof.

266. Corollary 3. - By drawing tangents at the vertices of regular inscribed polygons of 5, 10, 20, 40, etc., sides, regular polygons of the same numbers of sides may be circumscribed about a circle.

Give proof.

267. Construction. - Inscribe a regular polygon of 15 sid in a given circle.

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Required to inscribe in the circle a regular polygon of 15 sides.

Construction. 1. Draw AB, the side of a regular inscribed lecagon, and AC, the side of a regular inscribed hexagon, and draw BC.

2. Then BC is the side of the required polygon. Suggestion. AC is what part of the circle? AB is what

part of the circle? Then what part of the circle is BC?

268. Corollary 1. - Regular polygons of 15, 30, 60, etc., sides may be inscribed in a circle.

How? Give proof.

269. Corollary 2. By drawing tangents at the vertices of regular inscribed polygons of 15, 30, 60, etc., sides, regular polygons of the same numbers of sides may be circumscribed about a circle.

The proof is left to the student.

270. Summary. - The methods of constructing regular polygons of 3, 6, 12, etc., or 4, 8, 16, etc., or 5, 10, 20, etc., or 15, 30, 60, etc., sides by use of circles have been shown

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