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REGULAR POLYGONS. MEASUREMENT OF THE
PROPOSITION I. THEOREM (382'.) DEF. A regular polygon is one which is equiangular and equilateral.
383. A circle can be circumscribed about any regular polygon.
Hyp. ABCDE is a regular polygon.
Proof. Construct a circumference through A, B, and C, and
(Why ?) ZOBC= OCB.
(Why?) .. ZOBA = L OCD. But
OC = OB and AB= CD. (Why ?)
(Why?) .. OD= 0 A.
.. the circumference passes through D.
In like manner, it may be proven that the circumference passes through the remaining vertices of the polygon.
.. a circle can be circumscribed about the given polygon. Q.E.D.
PROPOSITION II. THEOREM 384. 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.
385. DEF. The center of a regular polygon is the common center of the circumscribed and inscribed circles of the polygon.
386. DEF. The radius of a regular polygon is the radius of the circumscribed circle.
387. DEF. The angle at the center is the angle between two radii drawn to the ends of a side.
388. DEF. The apothem of the polygon is the radius of the inscribed circle.
389. Cor. 1. The angle at the center of a regular polygon of n sides is equal to 4 right angles.
390. Cor. 2. The angle formed by an apothem to a side of a regular polygon, and a radius to an extremity of that side, is equal to ? right angles.
Ex. 915. The lines joining opposite vertices of a regular hexagon pass through the center.
Ex. 916. A triangle is regular if the centers of the circumscribed and inscribed circles coincide.
Ex. 917. A polygon is regular if the centers of the circumscribed and inscribed circles coincide.
Ex. 918. The angle at the center of a regular polygon is the supplement of an angle of the polygon.
PROPOSITION III. THEOREM 391. If the circumference of a circle is divided into any number of equal parts :
(1) The chords joining the points of division successively form a regular inscribed polygon.
(2) Tangents drawn at the points of division form a regular circumscribed polygon.
Hyp. The circumference ACE is divided into the equal arcs AB, BC, CD, etc.
(1) To prove ABCDE is a regular polygon. Proof. arc AB = arc BC = arc CD, etc. (Why?)
arc AC = arc BD = arc CE, etc. (Ax. 2.)
.. ZEAB = LABC= ZBCD, etc. (Why ?) But AB = BC = CD.
(Why?) .: polygon ABCDE is regular. (Why ?) (2) To prove tangents drawn at A, B, C, etc., form the regular circumscribed polygon FGHIK.
Proof. Z GAB= Z GBA= Z CBH = Z HCB, etc., (Why ?) and AB =BC = CD, etc.
(Why ?) .. A ABG, CBH, CDI, etc., are equal and isosceles. (Why?)
.:ZG= ZH= ZI, etc., and
AG = GB = BH = HC, etc. . Whence GH = HI = IK, etc.
(Ax. 2.) .. circumscribed polygon FGHIK is regular.
Q.E.D. 392. Cor. 1. The perimeter of an inscribed polygon is less than the perimeter of an inscribed polygon of double the number of sides.
393. Cor. 2. The perimeter of a circumscribed polygon is greater than the perimeter of a circumscribed polygon of double the number of sides.
Ex. 919. An equilateral polygon inscribed in a circle is regular.
Ex. 920. An equilateral polygon circumscribed about a circle is regular if the number of its sides is odd.
Ex. 921. An equiangular polygon circumscribed about a circle is regular.
PROPOSITION IV. PROBLEM 394. To inscribe a square in a given circle.
Construction. In the given circle ABC, draw diameters AC and BD perpendicular to each other, and join AB, BC, CD, and DA.
Then ABCD is the required square. [The proof is left to the student.]
395. 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 16, 32, ..., 2 sides may be constructed.
396. Cor. 2. By drawing tangents at A, B, C, and D, a square may be circumscribed about the circle.
Ex. 922. To circumscribe an octagon about a given circle.
Ex. 924. The side of an inscribed square is equal to the radius · multiplied by V2.
Ex. 925. Find the area of a square, if its radius is equal to r.
PROPOSITION V. PROBLEM 397. To inscribe a regular hexagon in a given circle.
Construction. In the given circle ACD, draw the radius AO.
From A as a center, with a radius equal to 0 A, draw an arc meeting the circumference in B. Draw AB.
AB is the side of a regular hexagon.
By applying the radius six times as a chord, the regular hexagon ABCDEF is formed.