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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

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, ÂŁ, 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
ÂŁ 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 ÂŁ, 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.

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.

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

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

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