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1*4 BOOK IV,^CIRCLES,

SECTION v.â€” TWO CIRCLES.

Proposition XXI.

d77. Tlieorein : If the distance between the centers

of two circles is equal to the sum of their radii, the

drdes are tangent extemaliy.

Btattftent : Let / 2, the distance between the centers of

any two circles, be equal to the sum of the radii. The cir-

cles arc tangent externally.

Consttactioii : Draw the straight line f 3 2, the parts / 3

and 3 2 being the radii. If the circumferences can have any

other point than 3 in common, let 4 be that point ; and draw

14 and 24.

Demoturtxation : The point 3 is obviously common to the

two circumferences. If 4 is also common, / 4 and 2 4 are

radii of the two circles ; and their sum is greater than / 2.

(Boor ii., Prop, xi.) This, however, is contrary to the hy-

pothesis. The circumferences have, then, but one point in

common, and are, therefore, tangent ; and, as each center

lies without the other circle, they are tangent externally.

Condnnon : The circles whose centers are / and 2, etc.

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SECTION V.^TWO CIRCLES, 185

Proposition XXII.

37Â§. Theorem : If the distance between the centers

of two circles is equal to the difference of the radii,

the circles are tangent internally*

Statement: Let / 2, the distance between the centers of any

two circles, equal the difference of the radii. The circles are

tangent internally.

Construction : Draw / 2, and produce it to 3, a point in

the circumference of the circle whose center is 2.

Demonstration : U 1 2, the difference of the radii, be added

to 2 3, the radius of one of the circles, the sum, the straight

line 12 3, is equal to the radius of the other circle. The

circumferences, have, then, one point, 3, in common. If

they can have another point in common, let 4 be that point.

/ 4 and 2 4 are, then, the radii of the two circles, and their

difference is less than the line / 2. (Book ii., Prop, xi.)

This, however, is contrary to the hypothesis. The circum-

ferences have, then, but one point in common, and they are,

therefore, tangent. As the center of one circle lies within

the other circle, they are tangent internally.

Conclusion : As the circles whose centers are / and 2 are, etc.

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t86 book IV.-^CIRCLES.

Proposition XXIII.

870, Theorem : If the circumferences of two circles

intersect^ the distance between the centers is less than

the sum and greater than the difference of the radii.

Statement : Let the circles whose centers are / and 2 bÂ«

any two circles whose circumferences intersect, as at 3 and

4. Prove that the distance between the centers, / and 2, is

less than the sum and greater than the difference of the

radii.

Construction : Draw the radii / 3 and 2 3.

Demonstration : The distance between the centers, / 2, is

one side of the triangle 13 2. It is, then, less than the sum

and greater than the difference between / 3 and 2 3, the

radii of the two circles. (Book ii., Prop, xi.)

Conclusion : The circles whose centers are / and 2, etc.

Corollary I. If the circumferences of two circles intersect,

the straight line joining the centers bisects at right angles

the straight line joining the points of intersection.

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SECTION v.â€” TWO CIRCLES, 187

Proposition XXIV.

2S0. Theorem : A circle ^nay be inscribed in, or cir-

cumscribed about f any regular polygon*

Statement : Let A BC H be a portion of any regular poly-

gon. A circle may be inscribed in, or circumscribed about

the polygon.

Gonstmction : Bisect two of the sides, as A B and B C, by per-

pendiculars meeting at (?. (Prop. XV.) From (? draw straight

lines to the extremities and middle points of the other sides.

Demonstration: All the small triangles into which the

regular polygon is divided are equal. (Book ii., Props, i.

and IV ) The lines from to the vertices are, then, equal ; and,

the lines from bisecting the sides are equal perpendiculars.

If, with as a center and the distance to any vertex as a

radius, a circumference be described, it will pass through all

the other vertices and be circumscribed about the polygon ;

or if, with as a. center and the distance to the middle point

of any side as a radius, a circumference be described, it will

pass through the middle points of all the other sides ; and,

as the sides are then perpendicular to radii at their outer

extremities, they are tangents to the circle (Book iv.. Prop.

xiii.), and the circle is inscribed within the regular polygon.

Conclusion : As A B CM is a portion of any, etc.

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i88

BOOK IV.^CIRCLES.

Proposition XXV.

dSl. Theorem : If a regular polygon of any num-

ber of sides he inscribed in a circle^ a regular polygon

of the same number of sides m,ay be circumscribed

about the circle ; or^ if a regular polygon of any num-

ber of sides be circumscribed about a circle, a regular

polygon of the same number of sides m^y be inscribed

in the circle.

Statement : Let A B C H be a portion of any regular poly-

gon inscribed . in, or circumscribed about the circle whose

center is 0. A regular polygon of the same number of sides

may be circumscribed about the circle to correspond with

the given inscribed polygon, or inscribed in the circle to

correspond with the given circumscribed polygon.

Construction : Draw A, B, C, and H to the vertices ;

also, the radii, Â£, OF, and OS, perpendicular to the sides

of the given polygon. (Prop, xiii.) To obtain the corre-

sponding circumscribed polygon, draw ab, be, and c h per-

pendicular to the radii E, OF, and S dX their outer ex-

tremities ; to obtain the corresponding inscribed pol)"gon,

draw ab, be, and c ^ so as to join the points where the lines

drawn to the vertices ; â€” viz., A, OB, C. and OH â€” intersect

the circumference.

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SECTION V.^TWO CIRCLES. 189

DemonstFation L : The sides of the given inscribed poly*

gon being equal chords, they are at equal distances IfOfit th6

center. (Prop, iv.) The small triangles into whkh thd

given inscribed polygon is divided are, then, alt equal.

(Book ii , Prop, i.) The small angles at the center, as /, 7,

3, etc., are thus all proved to be equal ; and, the triangles

aOE, bOE, bOF, cOF, cOS, and h S are all equal. (Book

II., Prop, ii.) The figure abch, etc., is, then^ a regular poly-

gon, as alt of its sides and angles are equal. The a0^^ fit

Â£ being equal, each is a right angle, and a b is, theA, a ikA*

gent. (Prop, xiii.) Similarly, 6 c and c h are shoii^rrt to be

tangents. The figure abc h, etc., is, then, a regular circum-

scribed polygon, and it has the same number of sides as the

given polygon.

Demoimtration II. : The sides of the given circumscribed

polygon being tangents, the angles at Â£, F, and S are tight

angles. (Prop, xiii.) The angles at the vertices of the

given polygon have been bisected, by constfuctiort ; thus,

the angles at B are equal. (Book ii.. Prop, iv.)

The series of triangles, A B, B C, COM, etc., are, then,

all equal. (Book ii.. Prop, i.) The angles at the center,

AO B, BO C, C H, etc., are, then, equal.

The series of triangles a Ob, bOc, cOh, etc., are, then,

equal, as their sides are equal radii. (Book ii.. Prop, i.)

The figure abch, etc., is, then, a regular polygon, is all df

its sides and angles are equal. It is an inscribed polygori^

as the sides are chords of the circle, and they are, mOrftOfVef',

the same in number as the sides of the given circumscribed

polygon.

Conclnnon : ABCH being a portion of any regular poly-

gon, it follows, etc.

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190 BOOK IV,â€” CIRCLES,

MISCELLANEOUS EXERCISES.

Problems.

3Â§2. L I. With a given radius, describe a circle tangent

to a given straight line and given circle.

2 With a given radius, describe a circle tangent to two

given circles.

3. Construct three circles having equal diameters, and

being tangent to one another.

II. 4. Construct a circle which shall touch two given

straight lines, and pass through a given point between them.

5. Describe a circle which shall pass through a given

point and be tangent to two given circles.

6. Describe a circle which shall be tangent to two given

straight lines and to a given circle.

7. Draw two concentric circles, such that the chords of

the outer circle tangent to the inner circle may be equal

to the diameter of the inner circle.

III. 8. Inscribe a circle in a given sector.

9 From the vertices of a triangle as centers, describe

three circles which shall touch each other, two and two.

10. Draw a straight line cutting two concentric circles,

so that the part intercepted by the circumference of the

greater may be double the part intercepted by the less.

Theorems.

2Â§3. I. II. If a straight line is a tangent to the interior

of two concentric circles, and a chord in the outer, it will

be bisected at the point of contact.

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MISCELLANEOUS EXERCISES, 19^

12. If a circle is described on the radius of another circle

as a diameter, any chord of the greater, passing through the

point of contact of the circles, is bisected by the circumfer-

ence of the smaller.

13. Prove that the shortest distance between two circum-

ferences is measured on the line which joins the centers.

II. 14. If two circles cut each other, and from either point

of intersection diameters be drawn, their extremities and

the other point of intersection are colli near.

15. If a straight line is drawn through the point of contact

of two circles which touch each other, terminating in their

circumferences, the radii drawn to its extremities are parallel ;

and, also, the tangents.

16. If two circles intersect, their common chord, pro-

duced, bisects the parts of their common tangents included

between the points of contact.

17. If the sides of an equiangular and equilateral pentagon

be produced to meet, the angles formed by these lines are

equal, and their sum is equal to two right angles.

18. The three common chords of three circles which inter-

sect each other, two and two, meet each other in one point.

III. 19. The square and the rhombus are the only paral-

lelograms in which a circle can be inscribed.

20. The straight line bisecting any angle of a triangle cuts

the. circumference of the circumscribing circle in a point

which is equidistant from the extremities of the opposite

side and from the center of the inscribed circle.

21. The entire plane space about a point can be filled,

without leaving vacant intervals, by equal equilateral tri-

angles, by equal squares, and by equal regular hexagons.

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192 BOOK IV.^CIRCLES.

Loci.

2Â§4. Find the following loci : I. 22. The points at a given

distance from a given point.

23. The points at a given distance from a given straight

line.

24. ThÂ« points equally distant from two parallel straight

lines.

25. The points equally distant from the circumferences

of two concentric circles.

26. The points equally distant from two intersecting

straight lines.

27. The vertex of a right-angled triangle with a given

hypotenuse.

28. The vertex of a triangle with a given base and a given

opposite angle.

29. The middle points of equal chords of a given length.

30. The middle points of secants from a given point to a

given circle.

31. The centers of circles tangent to two intersecting

straight lines.

32. The points, any one of which is equidistant from the

circumferences of two equal circles.

33. The points, the sum of the squares of the distances of

any point of which, from the four sides of a rectangle, shall

be equal to a given square.

II. 34. The middle points- of the chords of a circle pass-

ing through a given point (a) within the circle ; (b) on the

circumference ; (c) beyond the circumference.

35. The middle points of chords parallel to a given

straight line.

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MISCELLANEOUS EXERCISES. 193

36. The points of intersection of tangents including a

given angle.

37. The centers of circles tangent to a given straight line

at a given point.

38. The centers of circles tangent to a given circle at

a given point.

39. The middle point of a straight line moving between

the sides of a right angle.

III. 40. Center of circle with given radius and passing

through a given point.

41. Center of circle with given radius and tangent to a

given straight line.

Maxima and Minima.

aÂ§5. 42. Of all straight lines which can be drawn from

two given points to meet on the convex circumference of a

given circle, the sum of those two which make equal angles

with the tangent at the point of meeting will be the least.

43. Of polygons that are isoperimetric (i.e., that have

equal perimeters), and have the same number of sides, the

maximum is a regular polygon. (Isosceles triangles.)

44. Of isoperimetric regular polygons, the maximum is

that which has the greatest number of sides.

If one regular polygon has one side more than the other, this other

may be divided into two parts, placed to make an equivalent

irregular polygon of the same number of sides as given polygon.

45. Of equivalent regular polygons, that one has the least

perimeter which has the greatest number of sides.

Compare each with a regular polygon constructed with less number

of sides, but area equal to polygon with greater number of sides.

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194 BOOK IV.â€” CIRCLES,

46. The circumference of a circle incloses a greater area

than any polygon of equal perimeter. (Method of limits.)

47. The perimeter of a circle is less than the perimeter of

any equivalent polygon.

Numerical Exercises.

286. 48. Two angles of a triangle are, respectively,

41Â° 5' 31" and 37Â° 52' 49". Find the third angle.

49. How many degrees in each angle of a square ?

50. How many degrees in angle of equilateral triangle ?

51. How many degrees in each of the equal angles of an

isosceles triangle, the angle at the vertex being 45Â° 45' 40" ?

52. Which angle, if any, is obtuse in the triangle whose

sides are 3, 5, and 7 inches long ? Whose sides are 7, ti, 15 ?

53. What kind of a triangle could have sides, respectively,

5, TO, and 15 ? 6, 12, and 24 ?

54. Find the projections on the hypotenuse of the sides, 7

and 14, of a right angled triangle. Find the distance to the

hypotenuse from the opposite angle.

55. ThQ radius of a circle is 10. How far from the cen-

ter is a chord, 5 ?

56. The radius of a circle is 5. How far from the center

is a chord, 8 ?

57. A chord is 7 and its distance from the center is 3.

What is the radius ?

58. A chord is 9 and the diameter is to. What is the dis-

tance of the chord from the center ?

59. If a chord, 16 inches long, is 10 inches from the center,

how far from the center is a chord, 22 inches long ?

60. A circle has a radius, 9 ; what is the length of the

maximum chord through a point 7 inches from the center ?

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MISCELLANEOUS EXERCISES.

195

CONSTRUCTIONS.

3Â§7. Prove the following constructions :

Note. â€” Prove the problems in Arts. 48,49, 50, 60, 84, 85, 86. 121,

and 186.

7. A line through a given pointy parallel to a given straight

line,

5~

< ^ â€”

Jl ^ \ B A

2. A parallelogram^ two sideSy and included angle given.

J. A triangle, two angles, and included side given.

A I. 5/7 V

4. The third angle of a triangle, two angles given.

J. A triangle, two sides and angle opposite one given.

A-

^L

6. A right-angled triangle, hypotenuse, and one side given,

7. A triangle equivalent to a parallelogram.

8. A triangle equivalent to any polygon.

A triangle or parallelogram with given angle may, then, readily be

constructed equivalent to any given polygon.

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196 BOOK IV,â€” CIRCLES,

GENERAL SCHOLIA.

CONCYCLIC POINTS. .

2SÂ§, Conoyolic points are points that lie in the circum-

ference of the same circle.

Exercises, i. Any three points are concyclic. ^

2. The vertices of any regular polygon are concyclic.

3. The vertices of a quadrilateral are concyclic, if any two

opposite angles are supplementary.

4. The vertices of equal given vertical angle of triangles

on a given base are concyclic.

INSCRIBED AND ESCRIBED CIRCLES.

aÂ§9. An inscribed circle is a circle tangent to the three

sides of the triangle ; as / 2 5.

290. An escribed circle is a circle tangent to one side of a

triangle and to two other sides produced ; as 4 5 5.

291. Prove the following propositions with respect to the

inscribed and any escribed circle :

1. Tangents from the same point to a circle are equal.

2. Any tangent being a side with its production is equal to

half the perimeter of the triangle ; as k 5.

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GENERAL SCHOLIA. 197

3. Any tangent being an internal segment of a side is

equal to half of the perimeter diminished by the side opposite

the vertex from which the tangent is drawn ; as >l /, or, IB.

4. The intercepts between the points of contact made by

any two sides of the triangle produced are equal to each

other and to the third side of the triangle.

THE TRIANGLE AND NINE OP ITS CIRCLES.

â– '.V

The thn other six, by

centers. Prove the coDstruction of the nine circles. Find three

other sets of concyclic points.

Circles. Centers. Radii.

Circumscribed, / / >l

Inscribed, 2 Perp. from 7

Nine Points, 3 3 4

Pedal (3), 4, Bf6 4 0.50,60

C is the centroid ; and 0, the orthocenter ; 13^^ 30

393. Prove the following propositions :

1. The circumscribed circle bisects the straight lines, join-

ing the center of the inscribed circle with the centers of the

escribed circles.

2. Each vertex of the triangle is collinear with the centers

of two of the escribed circles.

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198 BOOK TV.â€”CIRCLES.

3. The center of the inscribed circle is collinear with the

center of any escribed circle and the opposite vertex.

4. Each center of the inscribed or the escribed circles is

the orthocenter of the triangle, having the other three centers

as its vertices. (See Â§ 293, for definition )

5. The four circles, each of which passes through three of

the centers of the escribed and inscribed circles, are equal.

6. The three circles, the circumference of each of which

passes through the extremities of any side of a triangle and

the orthocenter, equal one another and the original triangle.

Nineteen circles in all have been mentioned.

THE NINE-POINTS CIRCLE.

093. The orthocenter is the point at which the three per-

pendiculars from the vertices to the opposite sides meet.

294. The centroid of any triangle is the point at which

the medians of the triangle meet.

095. Prove the following propositions :

1. The middle points of the sides of a triangle are con-

cyclic with the feet of the perpendicular from the opposite

vertices, and the middle points of the lines joining the

orthocenter with the vertices. {Nine-paints circle.)

2. The center of the nine-points circle is the middle point

of the line joining the orthocenter and the center of the cir-

cumscribed circle.

3. The diameter of the nine-points circle is equal to the

radius of the circumscribed circle.

4. The orthocenter and the centroid are collinear with

the centers of the nine-points and circumscribed circles.

5. The nine-points circle is tangent to the inscribed and

escribed circles of a triangle.

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BOOK v.â€” PROPORTION.

SYNOPSIS.

SECTION I.â€” EQUAL RATIOS.

396. I. Definitions.

2 Propositions.

Prop. I. If four quantities are in proportion, the product of the

extremes is equal to the product of the means.

Prop. II. If four quantities are in proportion, they are in proportion

by alternation.

Prop. III. If four quantities are in proportion, they are in propor-

tion by inversion.

Prop. IV. If four quantities are in proportion, they are in propor-

tion by composition.

Prop. V. If four quantities are in proportion, they are in propor-

tion by division.

SECTION ILâ€” EQUAL PRODUCTS.

397. Prop. VI. If the product of two quantities is equal to the product

of two other quantities, the two quantities that form either

product may be made the extremes, and the other two quanti-

ties, the means, of a proportion.

Prop. VII. If the product of three quantities equals the product of

three other quantities, the three that form either product may

be separated in any manner to form the extremes, and the other

three quantities will form the means of a proportion.

Prop. VIII. If the product of two quantities is equal to the square

of a third quantity, the third quantity is a mean proportional

between th6 two other quantities.

199

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200 BOOK v.â€” PROPORTION.

Prop. IX. If the product of two quantities is equal to a third quan-

tity, the square root of the third quantity is a mean proportional

between the two other quantities.

Prop. X. If the product of three quantities is equal to a fourth

quantity, the square root of the fourth quantity is a mean pro-

portional between any one and the remaining two of the three

quantities.

SECTION III.â€” SEVERAL PROPORTIONS.

^^S. Prop. XI. If a couplet in each of two proportions is the same, the

other couplets form a proportion.

Prop. XII. If, in two proportions, the antecedents are the same, the

consequents form a proportion.

Prop. XIII. If, in two proportions, the consequents are the same,

the antecedents form a proportion.

Prop. XIV. If the terms of two proportions, taken in the same order,

are multiplied together, the products will form a proportion.

Prop. XV. If the terms of one proportion are divided by the terms

of another proportion, taken in the same order, the quotients

will form a proportion.

SECTION IV.â€” CONTINUED PROPORTIONS.

299. Prop. XVI. In any proportion, the sum of the antecedents is to

the sum of the consequents as any antecedent is to its consequent.

Prop. XVII. In any proportion, all the antecedents, or all the con-

sequents, may be multiplied by any quantity, and the results will

be proportional.

Prop. XVIII. In any proportion, all the antecedents, or all the con-

sequents, may be di^vided by any quantity, and the results will be

proportional.

Prop. XIX. The same powers of the terms of a proportion form a

proportion.

Prop. XX. The same roots of the terms of a proportion form a pro-

portion.

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SECTION I,â€” EQUAL RATIOS. 201

SECTION I.â€” EQUAL RATIOS.

DEFINITIONS.

300. Eatio is the relation between two quantities, ex-

pressed by division.

The ratio of 30 to 5 is -g-, ox 30 -k- 5 ; written in abbreviated form,

30 : 5, the ratio being equal to 6.

301. Proportion is an equality of ratios.

The ratio of 30 to 6 equals the ratio of 45 to 9, as each ratio is equal

to 5. The four quantities form a proportion, written thus :

30 45

6 = -9 ; or,

30 -i- 6 =^ 45 -^ 9 ; or,

30 : 6 = 45 : 9 ; or,

30 : 6 : : 45 : 9.

Any one of the preceding forms may be read : 30 is to 6 as 45 is to 9.

303. Proportion is the basis of mensuration. Either of

two quantities of the same kind may be taken as the meas-

ure of the other, and their relation may be expressed by

division, as a ratio ; and, when two quantities have a com-

mon measure, there is an equality of ratios, or a proportion.

Thus, if A= 5 X M, and, B = 7 x M, then, M = A, also,

B A B

^ = =-, and, ^ = ^ ; or, A : 5 : : B : 7.

7 o 7

303. The first and last terms are the extremes.

In the proportion, a : b : : c : d, the extremes are a and d,

304. The second and third terms are the means.

In the proportion, a : b : : c : d, the means are b and c.

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202 BOOK V.â€”PROPORTION,

305. The first, second, third, and fourth terms may be

called the first, second, third, and fourth proportionals.

When three quantities form a proportion, the second and third terms

being the same quantity, it is called a mean proportional.

306. Any two terms forming a ratio, as the first and

second terms, or the third and fourth terms, form a couplet.

307. The first terms in the different couplets are the

antecedents.

In the proportion, a : b : : c : d, the antecedents are a and c.

30Â§. The second terms in the different couplets are the

consequents.

In the proportion, a: b : : c : d, the consequents are b and d.

309. By alternation is meant comparison of the first term

with the third, and the second term with the fourth.

The proportion, a : b : : c : di by alternation becomes a : c : : b : d.

310. By inversion is meant making, in each couplet, the

1*4 BOOK IV,^CIRCLES,

SECTION v.â€” TWO CIRCLES.

Proposition XXI.

d77. Tlieorein : If the distance between the centers

of two circles is equal to the sum of their radii, the

drdes are tangent extemaliy.

Btattftent : Let / 2, the distance between the centers of

any two circles, be equal to the sum of the radii. The cir-

cles arc tangent externally.

Consttactioii : Draw the straight line f 3 2, the parts / 3

and 3 2 being the radii. If the circumferences can have any

other point than 3 in common, let 4 be that point ; and draw

14 and 24.

Demoturtxation : The point 3 is obviously common to the

two circumferences. If 4 is also common, / 4 and 2 4 are

radii of the two circles ; and their sum is greater than / 2.

(Boor ii., Prop, xi.) This, however, is contrary to the hy-

pothesis. The circumferences have, then, but one point in

common, and are, therefore, tangent ; and, as each center

lies without the other circle, they are tangent externally.

Condnnon : The circles whose centers are / and 2, etc.

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SECTION V.^TWO CIRCLES, 185

Proposition XXII.

37Â§. Theorem : If the distance between the centers

of two circles is equal to the difference of the radii,

the circles are tangent internally*

Statement: Let / 2, the distance between the centers of any

two circles, equal the difference of the radii. The circles are

tangent internally.

Construction : Draw / 2, and produce it to 3, a point in

the circumference of the circle whose center is 2.

Demonstration : U 1 2, the difference of the radii, be added

to 2 3, the radius of one of the circles, the sum, the straight

line 12 3, is equal to the radius of the other circle. The

circumferences, have, then, one point, 3, in common. If

they can have another point in common, let 4 be that point.

/ 4 and 2 4 are, then, the radii of the two circles, and their

difference is less than the line / 2. (Book ii., Prop, xi.)

This, however, is contrary to the hypothesis. The circum-

ferences have, then, but one point in common, and they are,

therefore, tangent. As the center of one circle lies within

the other circle, they are tangent internally.

Conclusion : As the circles whose centers are / and 2 are, etc.

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t86 book IV.-^CIRCLES.

Proposition XXIII.

870, Theorem : If the circumferences of two circles

intersect^ the distance between the centers is less than

the sum and greater than the difference of the radii.

Statement : Let the circles whose centers are / and 2 bÂ«

any two circles whose circumferences intersect, as at 3 and

4. Prove that the distance between the centers, / and 2, is

less than the sum and greater than the difference of the

radii.

Construction : Draw the radii / 3 and 2 3.

Demonstration : The distance between the centers, / 2, is

one side of the triangle 13 2. It is, then, less than the sum

and greater than the difference between / 3 and 2 3, the

radii of the two circles. (Book ii., Prop, xi.)

Conclusion : The circles whose centers are / and 2, etc.

Corollary I. If the circumferences of two circles intersect,

the straight line joining the centers bisects at right angles

the straight line joining the points of intersection.

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SECTION v.â€” TWO CIRCLES, 187

Proposition XXIV.

2S0. Theorem : A circle ^nay be inscribed in, or cir-

cumscribed about f any regular polygon*

Statement : Let A BC H be a portion of any regular poly-

gon. A circle may be inscribed in, or circumscribed about

the polygon.

Gonstmction : Bisect two of the sides, as A B and B C, by per-

pendiculars meeting at (?. (Prop. XV.) From (? draw straight

lines to the extremities and middle points of the other sides.

Demonstration: All the small triangles into which the

regular polygon is divided are equal. (Book ii., Props, i.

and IV ) The lines from to the vertices are, then, equal ; and,

the lines from bisecting the sides are equal perpendiculars.

If, with as a center and the distance to any vertex as a

radius, a circumference be described, it will pass through all

the other vertices and be circumscribed about the polygon ;

or if, with as a. center and the distance to the middle point

of any side as a radius, a circumference be described, it will

pass through the middle points of all the other sides ; and,

as the sides are then perpendicular to radii at their outer

extremities, they are tangents to the circle (Book iv.. Prop.

xiii.), and the circle is inscribed within the regular polygon.

Conclusion : As A B CM is a portion of any, etc.

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i88

BOOK IV.^CIRCLES.

Proposition XXV.

dSl. Theorem : If a regular polygon of any num-

ber of sides he inscribed in a circle^ a regular polygon

of the same number of sides m,ay be circumscribed

about the circle ; or^ if a regular polygon of any num-

ber of sides be circumscribed about a circle, a regular

polygon of the same number of sides m^y be inscribed

in the circle.

Statement : Let A B C H be a portion of any regular poly-

gon inscribed . in, or circumscribed about the circle whose

center is 0. A regular polygon of the same number of sides

may be circumscribed about the circle to correspond with

the given inscribed polygon, or inscribed in the circle to

correspond with the given circumscribed polygon.

Construction : Draw A, B, C, and H to the vertices ;

also, the radii, Â£, OF, and OS, perpendicular to the sides

of the given polygon. (Prop, xiii.) To obtain the corre-

sponding circumscribed polygon, draw ab, be, and c h per-

pendicular to the radii E, OF, and S dX their outer ex-

tremities ; to obtain the corresponding inscribed pol)"gon,

draw ab, be, and c ^ so as to join the points where the lines

drawn to the vertices ; â€” viz., A, OB, C. and OH â€” intersect

the circumference.

Digitized by

SECTION V.^TWO CIRCLES. 189

DemonstFation L : The sides of the given inscribed poly*

gon being equal chords, they are at equal distances IfOfit th6

center. (Prop, iv.) The small triangles into whkh thd

given inscribed polygon is divided are, then, alt equal.

(Book ii , Prop, i.) The small angles at the center, as /, 7,

3, etc., are thus all proved to be equal ; and, the triangles

aOE, bOE, bOF, cOF, cOS, and h S are all equal. (Book

II., Prop, ii.) The figure abch, etc., is, then^ a regular poly-

gon, as alt of its sides and angles are equal. The a0^^ fit

Â£ being equal, each is a right angle, and a b is, theA, a ikA*

gent. (Prop, xiii.) Similarly, 6 c and c h are shoii^rrt to be

tangents. The figure abc h, etc., is, then, a regular circum-

scribed polygon, and it has the same number of sides as the

given polygon.

Demoimtration II. : The sides of the given circumscribed

polygon being tangents, the angles at Â£, F, and S are tight

angles. (Prop, xiii.) The angles at the vertices of the

given polygon have been bisected, by constfuctiort ; thus,

the angles at B are equal. (Book ii.. Prop, iv.)

The series of triangles, A B, B C, COM, etc., are, then,

all equal. (Book ii.. Prop, i.) The angles at the center,

AO B, BO C, C H, etc., are, then, equal.

The series of triangles a Ob, bOc, cOh, etc., are, then,

equal, as their sides are equal radii. (Book ii.. Prop, i.)

The figure abch, etc., is, then, a regular polygon, is all df

its sides and angles are equal. It is an inscribed polygori^

as the sides are chords of the circle, and they are, mOrftOfVef',

the same in number as the sides of the given circumscribed

polygon.

Conclnnon : ABCH being a portion of any regular poly-

gon, it follows, etc.

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190 BOOK IV,â€” CIRCLES,

MISCELLANEOUS EXERCISES.

Problems.

3Â§2. L I. With a given radius, describe a circle tangent

to a given straight line and given circle.

2 With a given radius, describe a circle tangent to two

given circles.

3. Construct three circles having equal diameters, and

being tangent to one another.

II. 4. Construct a circle which shall touch two given

straight lines, and pass through a given point between them.

5. Describe a circle which shall pass through a given

point and be tangent to two given circles.

6. Describe a circle which shall be tangent to two given

straight lines and to a given circle.

7. Draw two concentric circles, such that the chords of

the outer circle tangent to the inner circle may be equal

to the diameter of the inner circle.

III. 8. Inscribe a circle in a given sector.

9 From the vertices of a triangle as centers, describe

three circles which shall touch each other, two and two.

10. Draw a straight line cutting two concentric circles,

so that the part intercepted by the circumference of the

greater may be double the part intercepted by the less.

Theorems.

2Â§3. I. II. If a straight line is a tangent to the interior

of two concentric circles, and a chord in the outer, it will

be bisected at the point of contact.

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MISCELLANEOUS EXERCISES, 19^

12. If a circle is described on the radius of another circle

as a diameter, any chord of the greater, passing through the

point of contact of the circles, is bisected by the circumfer-

ence of the smaller.

13. Prove that the shortest distance between two circum-

ferences is measured on the line which joins the centers.

II. 14. If two circles cut each other, and from either point

of intersection diameters be drawn, their extremities and

the other point of intersection are colli near.

15. If a straight line is drawn through the point of contact

of two circles which touch each other, terminating in their

circumferences, the radii drawn to its extremities are parallel ;

and, also, the tangents.

16. If two circles intersect, their common chord, pro-

duced, bisects the parts of their common tangents included

between the points of contact.

17. If the sides of an equiangular and equilateral pentagon

be produced to meet, the angles formed by these lines are

equal, and their sum is equal to two right angles.

18. The three common chords of three circles which inter-

sect each other, two and two, meet each other in one point.

III. 19. The square and the rhombus are the only paral-

lelograms in which a circle can be inscribed.

20. The straight line bisecting any angle of a triangle cuts

the. circumference of the circumscribing circle in a point

which is equidistant from the extremities of the opposite

side and from the center of the inscribed circle.

21. The entire plane space about a point can be filled,

without leaving vacant intervals, by equal equilateral tri-

angles, by equal squares, and by equal regular hexagons.

Digitized by

192 BOOK IV.^CIRCLES.

Loci.

2Â§4. Find the following loci : I. 22. The points at a given

distance from a given point.

23. The points at a given distance from a given straight

line.

24. ThÂ« points equally distant from two parallel straight

lines.

25. The points equally distant from the circumferences

of two concentric circles.

26. The points equally distant from two intersecting

straight lines.

27. The vertex of a right-angled triangle with a given

hypotenuse.

28. The vertex of a triangle with a given base and a given

opposite angle.

29. The middle points of equal chords of a given length.

30. The middle points of secants from a given point to a

given circle.

31. The centers of circles tangent to two intersecting

straight lines.

32. The points, any one of which is equidistant from the

circumferences of two equal circles.

33. The points, the sum of the squares of the distances of

any point of which, from the four sides of a rectangle, shall

be equal to a given square.

II. 34. The middle points- of the chords of a circle pass-

ing through a given point (a) within the circle ; (b) on the

circumference ; (c) beyond the circumference.

35. The middle points of chords parallel to a given

straight line.

Digitized by

MISCELLANEOUS EXERCISES. 193

36. The points of intersection of tangents including a

given angle.

37. The centers of circles tangent to a given straight line

at a given point.

38. The centers of circles tangent to a given circle at

a given point.

39. The middle point of a straight line moving between

the sides of a right angle.

III. 40. Center of circle with given radius and passing

through a given point.

41. Center of circle with given radius and tangent to a

given straight line.

Maxima and Minima.

aÂ§5. 42. Of all straight lines which can be drawn from

two given points to meet on the convex circumference of a

given circle, the sum of those two which make equal angles

with the tangent at the point of meeting will be the least.

43. Of polygons that are isoperimetric (i.e., that have

equal perimeters), and have the same number of sides, the

maximum is a regular polygon. (Isosceles triangles.)

44. Of isoperimetric regular polygons, the maximum is

that which has the greatest number of sides.

If one regular polygon has one side more than the other, this other

may be divided into two parts, placed to make an equivalent

irregular polygon of the same number of sides as given polygon.

45. Of equivalent regular polygons, that one has the least

perimeter which has the greatest number of sides.

Compare each with a regular polygon constructed with less number

of sides, but area equal to polygon with greater number of sides.

Digitized by

194 BOOK IV.â€” CIRCLES,

46. The circumference of a circle incloses a greater area

than any polygon of equal perimeter. (Method of limits.)

47. The perimeter of a circle is less than the perimeter of

any equivalent polygon.

Numerical Exercises.

286. 48. Two angles of a triangle are, respectively,

41Â° 5' 31" and 37Â° 52' 49". Find the third angle.

49. How many degrees in each angle of a square ?

50. How many degrees in angle of equilateral triangle ?

51. How many degrees in each of the equal angles of an

isosceles triangle, the angle at the vertex being 45Â° 45' 40" ?

52. Which angle, if any, is obtuse in the triangle whose

sides are 3, 5, and 7 inches long ? Whose sides are 7, ti, 15 ?

53. What kind of a triangle could have sides, respectively,

5, TO, and 15 ? 6, 12, and 24 ?

54. Find the projections on the hypotenuse of the sides, 7

and 14, of a right angled triangle. Find the distance to the

hypotenuse from the opposite angle.

55. ThQ radius of a circle is 10. How far from the cen-

ter is a chord, 5 ?

56. The radius of a circle is 5. How far from the center

is a chord, 8 ?

57. A chord is 7 and its distance from the center is 3.

What is the radius ?

58. A chord is 9 and the diameter is to. What is the dis-

tance of the chord from the center ?

59. If a chord, 16 inches long, is 10 inches from the center,

how far from the center is a chord, 22 inches long ?

60. A circle has a radius, 9 ; what is the length of the

maximum chord through a point 7 inches from the center ?

Digitized by

MISCELLANEOUS EXERCISES.

195

CONSTRUCTIONS.

3Â§7. Prove the following constructions :

Note. â€” Prove the problems in Arts. 48,49, 50, 60, 84, 85, 86. 121,

and 186.

7. A line through a given pointy parallel to a given straight

line,

5~

< ^ â€”

Jl ^ \ B A

2. A parallelogram^ two sideSy and included angle given.

J. A triangle, two angles, and included side given.

A I. 5/7 V

4. The third angle of a triangle, two angles given.

J. A triangle, two sides and angle opposite one given.

A-

^L

6. A right-angled triangle, hypotenuse, and one side given,

7. A triangle equivalent to a parallelogram.

8. A triangle equivalent to any polygon.

A triangle or parallelogram with given angle may, then, readily be

constructed equivalent to any given polygon.

Digitized by VjOOQIC

196 BOOK IV,â€” CIRCLES,

GENERAL SCHOLIA.

CONCYCLIC POINTS. .

2SÂ§, Conoyolic points are points that lie in the circum-

ference of the same circle.

Exercises, i. Any three points are concyclic. ^

2. The vertices of any regular polygon are concyclic.

3. The vertices of a quadrilateral are concyclic, if any two

opposite angles are supplementary.

4. The vertices of equal given vertical angle of triangles

on a given base are concyclic.

INSCRIBED AND ESCRIBED CIRCLES.

aÂ§9. An inscribed circle is a circle tangent to the three

sides of the triangle ; as / 2 5.

290. An escribed circle is a circle tangent to one side of a

triangle and to two other sides produced ; as 4 5 5.

291. Prove the following propositions with respect to the

inscribed and any escribed circle :

1. Tangents from the same point to a circle are equal.

2. Any tangent being a side with its production is equal to

half the perimeter of the triangle ; as k 5.

Digitized by

GENERAL SCHOLIA. 197

3. Any tangent being an internal segment of a side is

equal to half of the perimeter diminished by the side opposite

the vertex from which the tangent is drawn ; as >l /, or, IB.

4. The intercepts between the points of contact made by

any two sides of the triangle produced are equal to each

other and to the third side of the triangle.

THE TRIANGLE AND NINE OP ITS CIRCLES.

â– '.V

The thn other six, by

centers. Prove the coDstruction of the nine circles. Find three

other sets of concyclic points.

Circles. Centers. Radii.

Circumscribed, / / >l

Inscribed, 2 Perp. from 7

Nine Points, 3 3 4

Pedal (3), 4, Bf6 4 0.50,60

C is the centroid ; and 0, the orthocenter ; 13^^ 30

393. Prove the following propositions :

1. The circumscribed circle bisects the straight lines, join-

ing the center of the inscribed circle with the centers of the

escribed circles.

2. Each vertex of the triangle is collinear with the centers

of two of the escribed circles.

Digitized by

198 BOOK TV.â€”CIRCLES.

3. The center of the inscribed circle is collinear with the

center of any escribed circle and the opposite vertex.

4. Each center of the inscribed or the escribed circles is

the orthocenter of the triangle, having the other three centers

as its vertices. (See Â§ 293, for definition )

5. The four circles, each of which passes through three of

the centers of the escribed and inscribed circles, are equal.

6. The three circles, the circumference of each of which

passes through the extremities of any side of a triangle and

the orthocenter, equal one another and the original triangle.

Nineteen circles in all have been mentioned.

THE NINE-POINTS CIRCLE.

093. The orthocenter is the point at which the three per-

pendiculars from the vertices to the opposite sides meet.

294. The centroid of any triangle is the point at which

the medians of the triangle meet.

095. Prove the following propositions :

1. The middle points of the sides of a triangle are con-

cyclic with the feet of the perpendicular from the opposite

vertices, and the middle points of the lines joining the

orthocenter with the vertices. {Nine-paints circle.)

2. The center of the nine-points circle is the middle point

of the line joining the orthocenter and the center of the cir-

cumscribed circle.

3. The diameter of the nine-points circle is equal to the

radius of the circumscribed circle.

4. The orthocenter and the centroid are collinear with

the centers of the nine-points and circumscribed circles.

5. The nine-points circle is tangent to the inscribed and

escribed circles of a triangle.

Digitized by

BOOK v.â€” PROPORTION.

SYNOPSIS.

SECTION I.â€” EQUAL RATIOS.

396. I. Definitions.

2 Propositions.

Prop. I. If four quantities are in proportion, the product of the

extremes is equal to the product of the means.

Prop. II. If four quantities are in proportion, they are in proportion

by alternation.

Prop. III. If four quantities are in proportion, they are in propor-

tion by inversion.

Prop. IV. If four quantities are in proportion, they are in propor-

tion by composition.

Prop. V. If four quantities are in proportion, they are in propor-

tion by division.

SECTION ILâ€” EQUAL PRODUCTS.

397. Prop. VI. If the product of two quantities is equal to the product

of two other quantities, the two quantities that form either

product may be made the extremes, and the other two quanti-

ties, the means, of a proportion.

Prop. VII. If the product of three quantities equals the product of

three other quantities, the three that form either product may

be separated in any manner to form the extremes, and the other

three quantities will form the means of a proportion.

Prop. VIII. If the product of two quantities is equal to the square

of a third quantity, the third quantity is a mean proportional

between th6 two other quantities.

199

Digitized by

200 BOOK v.â€” PROPORTION.

Prop. IX. If the product of two quantities is equal to a third quan-

tity, the square root of the third quantity is a mean proportional

between the two other quantities.

Prop. X. If the product of three quantities is equal to a fourth

quantity, the square root of the fourth quantity is a mean pro-

portional between any one and the remaining two of the three

quantities.

SECTION III.â€” SEVERAL PROPORTIONS.

^^S. Prop. XI. If a couplet in each of two proportions is the same, the

other couplets form a proportion.

Prop. XII. If, in two proportions, the antecedents are the same, the

consequents form a proportion.

Prop. XIII. If, in two proportions, the consequents are the same,

the antecedents form a proportion.

Prop. XIV. If the terms of two proportions, taken in the same order,

are multiplied together, the products will form a proportion.

Prop. XV. If the terms of one proportion are divided by the terms

of another proportion, taken in the same order, the quotients

will form a proportion.

SECTION IV.â€” CONTINUED PROPORTIONS.

299. Prop. XVI. In any proportion, the sum of the antecedents is to

the sum of the consequents as any antecedent is to its consequent.

Prop. XVII. In any proportion, all the antecedents, or all the con-

sequents, may be multiplied by any quantity, and the results will

be proportional.

Prop. XVIII. In any proportion, all the antecedents, or all the con-

sequents, may be di^vided by any quantity, and the results will be

proportional.

Prop. XIX. The same powers of the terms of a proportion form a

proportion.

Prop. XX. The same roots of the terms of a proportion form a pro-

portion.

Digitized by

SECTION I,â€” EQUAL RATIOS. 201

SECTION I.â€” EQUAL RATIOS.

DEFINITIONS.

300. Eatio is the relation between two quantities, ex-

pressed by division.

The ratio of 30 to 5 is -g-, ox 30 -k- 5 ; written in abbreviated form,

30 : 5, the ratio being equal to 6.

301. Proportion is an equality of ratios.

The ratio of 30 to 6 equals the ratio of 45 to 9, as each ratio is equal

to 5. The four quantities form a proportion, written thus :

30 45

6 = -9 ; or,

30 -i- 6 =^ 45 -^ 9 ; or,

30 : 6 = 45 : 9 ; or,

30 : 6 : : 45 : 9.

Any one of the preceding forms may be read : 30 is to 6 as 45 is to 9.

303. Proportion is the basis of mensuration. Either of

two quantities of the same kind may be taken as the meas-

ure of the other, and their relation may be expressed by

division, as a ratio ; and, when two quantities have a com-

mon measure, there is an equality of ratios, or a proportion.

Thus, if A= 5 X M, and, B = 7 x M, then, M = A, also,

B A B

^ = =-, and, ^ = ^ ; or, A : 5 : : B : 7.

7 o 7

303. The first and last terms are the extremes.

In the proportion, a : b : : c : d, the extremes are a and d,

304. The second and third terms are the means.

In the proportion, a : b : : c : d, the means are b and c.

Digitized by

202 BOOK V.â€”PROPORTION,

305. The first, second, third, and fourth terms may be

called the first, second, third, and fourth proportionals.

When three quantities form a proportion, the second and third terms

being the same quantity, it is called a mean proportional.

306. Any two terms forming a ratio, as the first and

second terms, or the third and fourth terms, form a couplet.

307. The first terms in the different couplets are the

antecedents.

In the proportion, a : b : : c : d, the antecedents are a and c.

30Â§. The second terms in the different couplets are the

consequents.

In the proportion, a: b : : c : d, the consequents are b and d.

309. By alternation is meant comparison of the first term

with the third, and the second term with the fourth.

The proportion, a : b : : c : di by alternation becomes a : c : : b : d.

310. By inversion is meant making, in each couplet, the

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