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BOOK V

REGULAR POLYGONS. MEASUREMENT OF THE

CIRCLE

REGULAR POLYGONS

PROPOSITION I. THEOREM

388. DEF. A regular polygon is a polygon which is both equiangular and equilateral.

389. A circle can be circumscribed about any regular polygon.

[blocks in formation]

Given ABCDE, a regular polygon of n sides.

To prove that a circle can be circumscribed about ABCDE. Proof. Construct a circle through A, B, and C, and let o be its center.

[blocks in formation]
[blocks in formation]

In like manner, it may be proved that the circle passes through the remaining vertices of the polygon.

.. a circle can be circumscribed about the given polygon.

PROPOSITION II. THEOREM

Q. E. D.

390. A circle can be inscribed in any regular polygon.

HINT. Circumscribe a circle about the given polygon and prove that

the center is equidistant from the sides.

391. DEF. The center (0) of a regular polygon is the common center of the circumscribed and inscribed circles of the polygon.

392. DEF. The radius of a E regular polygon is the radius of the circumscribed circle; as OA.

393. DEF. The central angle is the angle between two radii drawn to the ends of one side; as AOB.

B

(197)

394. DEF. The apothem of the polygon is the radius of the inscribed circle; as OF.

395. COR. The central angle of a regular polygon of n

sides is equal to right angles.

4
n

Ex. 1282. If ABCDEF is a regular hexagon, then

LADCLAEC LAFC.

=

Ex. 1283. If two diagonals AC and BE of a regular polygon ABCDEFG intersect in Q, then AQ × QC = BQ × QE.

Ex. 1284. If in the regular heptagon ABCDEFG, the prolongations of AB and the diagonal of EC meet in H, then HB × HA : = HC × HE. Ex. 1285. Any central angle of a regular polygon is the supplement of an angle of the polygon.

Ex. 1286. A triangle is regular if the centers of the circumscribed and inscribed circles coincide.

Ex. 1287. A polygon is regular if the centers of the circumscribed and inscribed circles coincide.

[blocks in formation]

396. If the circumference of a circle is divided into any number of equal parts:

[blocks in formation]

(1) The chords joining the points of division succes

sively form a regular inscribed polygon.

(2) Tangents drawn at the points of division form a regular circumscribed polygon.

[blocks in formation]

(2) To prove tangents drawn at A, B, C, etc., form the regular circumscribed polygon FGHIK.

(Why?)

(Why?)

[blocks in formation]

.. ▲ ABG, CBH, CDI, etc., are congruent and isosceles. (Why?)

and

Whence

.. ≤ G = ≤ H = ▲ I, etc.

AG = GB = BH = HC, etc.
GH = HI = IK, etc.

.. circumscribed polygon FGHIK is regular.

(Ax. 2.)

Q. E. D.

397. COR. 1. The perimeter of a regular inscribed polygon is less than the perimeter of a regular inscribed polygon of double the number of sides.

398. COR. 2. The perimeter of a regular circumscribed polygon is greater than the perimeter of a regular circumscribed polygon of double the number of sides.

Ex. 1288. If a regular hexagon is inscribed in a circle, and the midpoints of the arcs which are subtended by the sides are joined to the vertices of the hexagon, a regular polygon of twelve sides is formed.

Ex. 1289. An equilateral polygon inscribed in a circle is regular. Ex. 1290. An equiangular polygon circumscribed about a circle is regular.

PROPOSITION IV. PROBLEM

399. To inscribe a square in a given circle.

Construction. In the given circle ABC, draw diameters AC and BD perpendicular to each other, and draw AB, BC, CD, and

DA.

Then ABCD is the required square.

[The proof is left to the student.]

400. COR. 1. By bisecting the central angles, the arcs AB, BC, etc., will be bisected, and a polygon of eight sides may be inscribed in the circle. By repeating the process, polygons of be constructed.

16, 32, ***,

B

2" sides

may

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