388. To construct a polygon similar to a given polygon and having a given ratio to it.

389. To construct a square equivalent to a given parallelogram.

390. Cor. 1. A square may be constructed equivalent to a given triangle, by taking for its side a mean proportional between the base and one-half the altitude of the triangle.

391. Cor. 2. A square may be constructed equivalent to a given polygon, by first reducing the polygon to an equivalent triangle, and then constructing a square equivalent to the triangle.

392. To construct a parallelogram equivalent to a given square and having the sum of its base and altitude equal to a given line.

393. To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.

394. To construct a polygon similar to a given polygon P, and equivalent to a given polygon Q.

BOOK V.

THEOREMS.

396. An equilateral polygon inscribed in a circle is a regular polygon.

397. A circle may be circumscribed about, and a circle may be inscribed in, any regular polygon.

402. Cor. 1. The angle at the centre of a regular polygon is equal to four right angles divided by the number of sides of the polygon.

403. Cor. 2. The radius drawn to any vertex of a regular polygon bisects the angle at the vertex.

404. Cor. 3. The interior angle of a regular polygon is the supplement of the angle at the centre.

405. If the circumference of a circle is divided into any number of equal parts, the chords joining the successive points of division form a regular inscribed polygon, and the tangents drawn at the points of division form a regular circumscribed polygon.

406. Cor. 1. Tangents to a circumference at the vertices of a regular inscribed polygon form a regular circumscribed polygon of the same number of sides.

407. Cor. 2. If a regular polygon is inscribed in a circle, the tangents drawn at the middle points of the arcs subtended by the sides of the polygon form a circumscribed regular polygon, whose sides are parallel to the sides of the inscribed polygon and whose vertices lie on the radii (prolonged) of the inscribed polygon.

408. Cor. 3. If the vertices of a regular inscribed polygon are joined to the middle points of the arcs subtended by the sides of the polygon, the joining lines form a regular inscribed polygon of double the number of sides.

409. Cor. 4. If tangents are drawn at the middle points of the arcs between adjacent points of contact of the sides of a regular circumscribed polygon, a regular circumscribed polygon of double the number of sides is formed.

411. Two regular polygons of the same number of sides are similar.

412. Cor. The areas of two regular polygons of the same number of sides are to each other as the squares of any two homologous sides.

413. The perimeters of two regular polygons of the same number of sides are to each other as the radii of their circumscribed circles, and also as the radii of their inscribed circles.

414. Cor. The areas of two regular polygons of the same number of sides are to each other as the squares of the radii of their circumscribed circles, and also as the squares of the radii of their inscribed circles.

415. The difference between the lengths of the perimeters of a regular inscribed polygon and of a similar circumscribed polygon is indefinitely diminished as the number of the sides of the polygons is indefinitely increased.

416. Cor. The difference between the areas of a regular inscribed polygon and of a similar circumscribed polygon is indefinitely diminished as the number of the sides of the polygons is indefinitely increased.

418. Two circumferences have the same ratio as their radii.

419. Cor. The ratio of a circumference of a circle to its diameter is constant.

421. The area of a regular polygon is equal to one-half the product of its apothem by its perimeter.

423. The area of a circle is equal to one-half the product of its radius by its circumference.

424. Cor. 1. The area of a sector equals one-half the product of its radius by its arc.

425. Cor. 2. The area of a circle equals times the square of its radius.

426. Cor. 3. The areas of two circles are to each other as the squares of their radii.

427. Cor. 4. Similar arcs, being like parts of their respective circumferences, are to each other as their radii; similar sectors, being like parts of their respective circles, are to each other as the squares of their radii.

428. The areas of two similar segments are to each other as the squares of their radii.

PROBLEMS OF CONSTRUCTION.

429. To inscribe a square in a given circle.

430. Cor. By bisecting the arcs subtended by the sides of a square, a regular polygon of eight sides may be inscribed in the circle; and, by continuing the process, regular polygons of sixteen, thirty-two, sixty-four, etc., sides may be inscribed.

431. To inscribe a regular hexagon in a given circle.

432. Cor. 1. By joining the alternate vertices of a regular inscribed hexagon, an equilateral triangle is inscribed in the circle.

433. Cor. 2. By bisecting the arcs of a regular inscribed hexagon, a regular polygon of twelve sides may be inscribed in the circle; and, by continuing the process, regular polygons of twenty-four, forty-eight, etc., sides may be inscribed.

434. To inscribe a regular decagon in a given circle. 435. Cor. 1. By joining the alternate vertices of a regular inscribed decagon, a regular pentagon is inscribed. 436. Cor. 2. By bisecting the arcs of a regular inscribed decagon, a regular polygon of twenty sides may be inscribed;

and by continuing the process, regular polygons of forty, eighty, etc., sides may be inscribed.

437. To inscribe in a given circle a regular pentedecagon, or polygon of fifteen sides.

438. Cor. By bisecting the arcs of a regular inscribed pentedecagon, a regular polygon of thirty sides may be inscribed; and, by continuing the process, regular polygons of sixty, one hundred and twenty, etc., sides may be inscribed.

439. To inscribe in a given circle a regular polygon similar to a given regular polygon.

440. Given the radius and the side of a regular inscribed polygon, to find the side of the regular inscribed polygon of double the number of sides.

441. To compute the ratio of the circumference of a circle to its diameter approximately.

THEOREMS.

445. Of all triangles having two given sides, that in which these sides include a right angle is the maximum.

446. Of all triangles having the same base and equal perimeters, the isosceles triangle is the maximum.

447. Of all polygons with sides all given but one, the maximum can be inscribed in a semicircle which has the undetermined side for its diameter.

448. Of all polygons with given sides, that which can be inscribed in a circle is the maximum.

449. Of isoperimetric polygons of the same number of sides, the maximum is equilateral.

450. Cor. The maximum of isoperimetric polygons of the same number of sides is a regular polygon.

451. Of isoperimetric regular polygons, that which has the greatest number of sides is the maximum.