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362. To construct a polygon similar to a given polygon, and having two and a half times its area.

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Let P be the given polygon.

It is required to construct a polygon similar to P, and equivalent to 2 P.

Let A B be a side of the given polygon P.

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(the homologous sides of similar polygons are to each other as the square roots

of their areas).

Take any convenient unit of length, as MC, and apply it

six times to the indefinite line MN.

On MO (= 3 M C) describe a semi-circumference;

and on M N (= 6 M C') describe a semi-circumference. At C erect a to MN, intersecting the semi-circumferences at D and H.

Then CD is the √2, and C H is the √5.
Draw CY, making any convenient with CH.

§ 360

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Then CY will equal x, and be a side of the polygon required, homologous to A B.

For

CD CH: CE

CY,

$275

(a line drawn through two sides of a ▲, to the third side, divides the two sides proportionally).

Substitute their equivalents for C D, CH, and CE;

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On CY, homologous to A B, construct a polygon similar to the given polygon P;

and this is the polygon required.

Q. E. F.

Ex. 1. The perpendicular distance between two parallels is 30, and a line is drawn across them at an angle of 45°; what is its length between the parallels?

2. Given an equilateral triangle each of whose sides is 20; find the altitude of the triangle, and its area.

3. Given the angle A of a triangle equal to of a right angle; the angle B equal to of a right angle, and the side a, opposite the angle A, equal to 10; construct the triangle.

4. The two segments of a chord intersected by another chord are 6 and 5, and one segment of the other chord is 3; what is the other segment of the latter chord?

5. If a circle be inscribed in a right triangle: show that the difference between the sum of the two sides containing the right angle and the hypotenuse is equal to the diameter of the circle.

6. Construct a parallelogram the area and perimeter of which shall be respectively equal to the area and perimeter of a given triangle.

7. Given the difference between the diagonal and side of a square; construct the square.

BOOK V.

REGULAR POLYGONS AND CIRCLES.

363. DEF. A Regular Polygon is a polygon which is equilateral and equiangular.

PROPOSITION I. THEOREM.

364. Every equilateral polygon inscribed in a circle is a regular polygon.

A

C

D

B

Let ABC, etc., be an equilateral polygon inscribed

in a circle.

We are to prove the polygon A B C, etc., regular.

The arcs A B, BC, C D, etc., are equal,
(in the same O, equal chords subtend equal arcs).

§ 182

.. arcs A B C, BC D, etc., are equal,

Ax. 6

.. the A, B, C, etc., are equal,
(being inscribed in equal segments).

.. the polygon ABC, etc., is a regular polygon, being

equilateral and equiangular.

Q. E. D.

PROPOSITION II. THEOREM.

365. I. A circle may be circumscribed about a regular polygon.

II. A circle may be inscribed in a regular polygon.

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may be circumscribed about this may be inscribed in this regular

Let ABCD, etc., be a regular polygon. We are to prove that a regular polygon, and also a O polygon.

CASE I. Describe a circumference passing through A, B, and C. From the centre O, draw O A, O D,

and draw Os to chord BC.

On Os as an axis revolve the quadrilateral O A Bs,

until it comes into the plane of Os CD.

The line s B will fall upon s C,

(for LOs B = 40s C, both being rt. ▲).

The point B will fall upon C,

(since s B

= 8 C).

The line BA will fall upon C D,

(since ▲ B = LC, being of a regular polygon).

The point A will fall upon D,

=

(since B A C D, being sides of a regular polygon).

.. the line OA will coincide with line O D,
(their extremities being the same points).

.. the circumference will pass through D.

§ 183

§ 363

$363

In like manner we may prove that the circumference, passing through vertices B, C, and D will also pass through the vertex E, and thus through all the vertices of the polygon in succession.

CASE II.The sides of the regular polygon, being equal chords of the circumscribed O, are equally distant from the centre, § 185

.. a circle described with the centre O and a radius Os will touch all the sides, and be inscribed in the polygon. § 174

Q E. D.

366. DEF. The Centre of a regular polygon is the common centre of the circumscribed and inscribed circles.

367. DEF. The Radius of a regular polygon is the radius OA of the circumscribed circle.

368. DEF. The Apothegm of a regular polygon is the radius. Os of the inscribed circle.

369. DEF. The Angle at the centre is the angle included by the radii drawn to the extremities of any side.

PROPOSITION III. THEOREM.

370. Each angle at the centre of a regular polygon is equal to four right angles divided by the number of sides of the polygon.

D

B

Let ABC, etc., be a regular polygon of n sides.

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371. COROLLARY. The radius drawn to any vertex of a

regular polygon bisects the angle at that vertex.

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