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388. The side of a circumscribed equilateral triangle is equal to twice the side of the similar inscribed triangle. Find the ratio of their areas. 389. The apothem of an inscribed equilateral triangle is equal to half the radius of the circle.

390. The apothem of an inscribed regular hexagon is equal to half the side of the inscribed equilateral triangle.

391. The area of an inscribed regular hexagon is equal to threefourths of that of the circumscribed regular hexagon.

392. The area of an inscribed regular hexagon is a mean proportional between the areas of the inscribed and the circumscribed equilateral triangles.

393. The area of an inscribed regular octagon is equal to that of a rectangle whose sides are equal to the sides of the inscribed and the circumscribed squares.

394. The area of an inscribed regular dodecagon is equal to three times the square of the radius.

395. Every equilateral polygon circumscribed about a circle is regular if it has an odd number of sides.

396. Every equiangular polygon inscribed in a circle is regular if it has an odd number of sides.

397. Every equiangular polygon circumscribed about a circle is regular.

398. Upon the six sides of a regular hexagon squares are constructed outwardly. Prove that the exterior vertices of these squares are the vertices of a regular dodecagon.

399. The alternate vertices of a regular hexagon are joined by straight lines. Prove that another regular hexagon is thereby formed. Find the ratio of the areas of the two hexagons.

400. The radius of an inscribed regular polygon is the mean proportional between its apothem and the radius of the similar circumscribed regular polygon.

401. The area of a circular ring is equal to that of a circle whose diameter is a chord of the outer circle and a tangent to the inner circle.

402. The square of the side of an inscribed regular pentagon is equal to the sum of the squares of the radius of the circle and the side of the

If R denotes the radius of a circle, and a one side of a regular inscribe polygon, show that:

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406. If on the legs of a right triangle, as diameters, semicircles a described external to the triangle, and from the whole figure a semicirc on the hypotenuse is subtracted, the remainder is equivalent to the give triangle,

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409. If the radius of a circle is r, and the side of an inscribed regul polygon is a, show that the side of the similar circumscribed regul polygon is equal to

2 ar

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410. The radius of a circle = r. Prove that the area of the inscrib regular octagon is equal to 2r2√2.

411. The sides of three regular octagons are 3 feet, 4 feet, and 5 fe respectively. Find the side of a regular octagon equal in area to t sum of the areas of the three given octagons.

412. What is the width of the ring between two concentric circu ferences whose lengths are 440 feet and 330 feet?

413. Find the angle subtended at the centre by an arc 6 feet 5 inch long, if the radius of the circle is 8 feet 2 inches.

414. Find the angle subtended at the centre of a circle by an a whose length is equal to the radius of the circle.

415. What is the length of the arc subtended by one side of a regu dodecagon inscribed in a circle whose radius is 14 feet?

416. Find the side of a square equivalent to a circle whose radius 56 feet.

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417. Find the area of a circle inscribed in a square containing 196 square feet.

418. The diameter of a circular grass plot is 28 feet. Find the diameter of a circular plot just twice as large.

419. Find the side of the largest square that can be cut out of a circular piece of wood whose radius is 1 foot 8 inches.

420. The radius of a circle is 3 feet. What is the radius of a circle 25 times as large? as large? as large?

421. The radius of a circle is 9 feet. What are the radii of the concentric circumferences that will divide the circle into three equivalent parts?

422. The chord of half an arc is 12 feet, and the radius of the circle is 18 feet. Find the height of the arc.

423. The chord of an arc is 24 inches, and the height of the arc is 9 inches. Find the diameter of the circle.

424. Find the area of a sector, if the radius of the circle is 28 feet, and the angle at the centre 221o.

425. The radius of a circle = r. Find the area of the segment subtended by one side of the inscribed regular hexagon.

426. Three equal circles are described, each touching the other two. If the common radius is r, find the area contained between the circles.

PROBLEMS.

To circumscribe about a given circle :

427. An equilateral triangle.

428. A square.

429. A regular hexagon.

430. A regular octagon.

431. To draw through a given point a line so that it shall divide a given circumference into two parts having the ratio 3: 7.

432. To construct a circumference equal to the sum of two given circumferences.

433. To construct a circle equivalent to the sum of two given circles. 434. To construct a circle equivalent to three times a given circle. 435. To construct a circle equivalent to three-fourths of a given circle. To divide a given circle by a concentric circumference:

436. Into two equivalent parts. 437. Into five equivalent parts.

THEOREMS.

438. The line joining the feet of the perpendiculars dropped from th extremities of the base of an isosceles triangle to the opposite sides i parallel to the base.

439. If AD bisect the angle A of a triangle ABC, and BD bisect th exterior angle CBF, then angle ADB equals one-half angle ACB.

440. The sum of the acute angles at the vertices of a pentagram (five pointed star) is equal to two right angles.

441. The bisectors of the angles of a parallelogram form a rectangle

442. The altitudes AD, BE, CF of the triangle ABC bisect the angle of the triangle DEF.

HINT. Circles with AB, BC, AC as diameters will pass through E an D, E and F, D and F, respectively.

443. The portions of any straight line intercepted between the ci cumferences of two concentric circles are equal.

444. Two circles are tangent internally at P, and a chord AB of th larger circle touches the smaller circle at C. Prove that PC bisects th angle APB.

HINT. Draw a common tangent at P, and apply ? 263, 269, 145. 445. The diagonals of a trapezoid divide each other into segmen reciprocally proportional.

446. The perpendiculars from two vertices of a triangle upon th opposite sides divide each other into segments reciprocally proportiona

447. If through a point P in the circumference of a circle two chor are drawn, the chords and the segments between P and a chord parall to the tangent at Pare reciprocally proportional.

448. The perpendicular from any point of a circumference upon chord is a mean proportional between the perpendiculars from the sar point upon the tangents drawn at the extremities of the chord.

449. In an isosceles right triangle either leg is a mean proportion between the hypotenuse and the perpendicular upon it from the vert of the right angle.

450. The area of a triangle is equal to half the product of its peri eter by the radius of the inscribed circle.

451. The perimeter of a triangle is to one side as the perpendicular from the opposite vertex is to the radius of the inscribed circle.

452. The sum of the perpendiculars from any point within a convex equilateral polygon upon the sides is constant.

453. A diameter of a circle is divided into any two parts, and upon these parts as diameters semi-circumferences are described on opposite sides of the given diameter. Prove that the sum of their lengths is equal to the semi-circumference of the given circle, and that they divide the circle into two parts whose areas have the same ratio as the two parts into which the diameter is divided.

454. Lines drawn from one vertex of a parallelogram to the middle points of the opposite sides trisect one of the diagonals.

455. If two circles intersect in the points A and B, and through A any secant CAD is drawn limited by the circumferences at C and D, the straight lines BC, BD, are to each other as the diameters of the circles.

456. If three straight lines AA', BB', CC', drawn from the vertices of a triangle ABC to the opposite sides, pass through a common point within the triangle, then

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457. Two diagonals of a regular pentagon, not drawn from a common vertex, divide each other in extreme and mean ratio.

Loci.

458. Find the locus of a point P whose distances from two given points A and B are in a given ratio (m: n).

459. From a point A in a given circumference a straight line AP of fixed length is drawn parallel to a given line BC. Find the locus of P.

460. From a fixed point A a straight line AB is drawn to any point in a given straight line CD, and then divided at P in a given ratio (mn). Find the locus of the point P.

461. Find the locus of a point whose distances from two given straight lines are in a given ratio. (The locus consists of two straight lines.)

462. Find the locus of a point the sum of whose distances from two

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