Euclid. # Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

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constant.

15. Point out clearly the difference in the proofs of the two latter cases

in Euc. VI. 7.

16. From the corollary of Euc. vi. 8, deduce a proof of Euc. i. 47.

17. Shew how the last two properties stated in Euc. vi. 8. Cor. may

be deduced from Euc. i. 47 ; ii. 2 ; vi. 17.

18. Given the nth part of a straight line, find by a Geometrical con-

struction, the (n + l)tli part.

19. Define what is meant by a mean proportional between two given

lines : and find a mean proportional between the lines whose lengths are

4 and 9 units resj)ectively. Is the method you employ suggested by any

Propositions in any of the first four books ?

20. Determine a third proportional to two lines of 5 and 7 units : and

a fourth proportional to three lines of o, 7, 9, units.

21. Find a straight line which shall have to a given straight line, the

ratio of 1 to Vo.

22. Define reciprocal figures. Enunciate the propositions proved re-

specting such figures in the Sixth Book.

23. Give the corollary, Euc. vi. 8, and prove thence that the Arith-

metic mean is greater than the Geometric between the same extremes.

24. If two equal triangles have two angles together equal to two

right angles, the sides about those angles are reciprocally proportional.

25. Give Algebraical proofs of Prop. 16 and 17 of Book vi.

26. Enunciate and prove the converse of Euc. vi. 15.

27. Explain what is meant by saying, that "similar triangles are in the

duplicate ratio of their homologous sides."

28. What are the data which determine triangles both in species and

magnitude ? How are those data expressed in Geometry ?

29. If the ratio of the homologous sides of two triangles be as 1 to

4, what is the ratio of the triangles ? And if the ratio of the triangles be

as 1 to 4, what is the ratio of the homologous sides ?

30. Shew that one of the triangles in the figure, Euc. iv. 10, is a mean

proportional between the other two.

31. What is the algebraical interpretation of Euc. vi. 19 ?

32. From j'-our definition of Proportion, prove that the diagonals of

a square are in the same proportion as their sides.

33. What propositions does Euclid prove respecting similar polygons ?

34. The parallelograms about the diameter of a parallelogram are similar

to the whole and to one another. Shew when they are equal. â–

35. Prove Algebraically, that the areas (1) of similar triangles and (2)

of similar parallelograms are proportional to the squares of their homo-

logous sides.

36. How is it shewn that equiangular parallelograms have to one

another the ratio which is compounded of the ratios of their bases and al-

titudes ?

37. To find two lines which shall have to each other, the ratio com-

pounded of the ratios of the lines A to B, and C to D.

38. State the force of the condition ** similarly described ;" and shew

that, on a given straight line, there may be described as many polygons

of different magnitudes, similar to a given polygon, as there are sides of

Afferent lengths in the polygon.

â€¢

QUESTIONS ON BOOK VF. oOl

39. Describe a triangle similar to a given triangle, and having its

area double that of the given triangle.

40. The three sides of a triangle are 7, 8, 9 units respectively; deter-

mine the length of the lines which meeting the base, and the base produced,

bisect the interior angle opposite to the greatest side of the triangle,

and the adjacent exterior angle.

41. The three sides of a triangle are 3, 4, 5 inches respectively ; find

the lengths of the external segments of the sides determined by the lines

which bisect the exterior angles of the triangle.

42. What are the segments into which the hypotenuse of a right-

angled triangle is divided by a perpendicular drawn from the right angle,

if the sides containing it are a and 3a units respectively ?

43. If the three sides of a triangle be 3, 4, 5 units respectively : what

are the parts into which they are divided by the lines which bisect the

angles opposite to them ?

44. If the homologous sides of two triangles be as 3 to 4, and the area

of one triangle be known to contain 100 square units ; how many square

units are contained in the area of the other triangle ?

45. Prove that if BD be taken in AB produced (fig. Euc. vi. 30)

equal to the greater segment AC^ then AD is divided in extreme and

mean ratio in the point B.

Shew also, that in the series 1, 1, 2, 3, 5, 8, &c. in which each term is

the sum of the two preceding terms, the last two terms perpetually ap-

proach to the proportion of the segments of a line divided in extreme and

mean ratio. Find a general expression (free from surds) for the Mth term

of this series.

46. The parts of a line divided in extreme and mean ratio are incom-

mensurable with each other.

47. Shew that in Euclid's figure (Euc. ii. 11.) four other lines, besides

the given line, are divided in the required manner.

48. Enunciate Euc. vi. 31. What theorem of a previous book is in-

cluded in this proposition ?

49. What is the superior limit, as to magnitude, of the angle at the

circumference in Euc. vi. 33 ? Shew that the proof may be extended by

withdrawing the usually supposed restriction as to angular magnitude ;

and then deduce, as a corollary, the proposition respecting the magnitudes

of angles in segments greater than, equal to, or less than a semicircle.

50. The sides of a triangle inscribed in a circle are a, 6, c, units respec-

tively : find by Euc. vi. c, the radius of the circumscribing circle.

51. Enunciate the converse of Euc. vi. d.

52. Shew independently that Euc. vi. d, is true when the quadri-

lateral figure is rectangular.

53. Shew that the rectangles contained by the opposite sides of a

quadrilateral figure which does not admit of having a circle described

about it, are together greater than the rectangle contained by the diagonals.

54. What diff'erent conditions may be stated as essential to the possi-

bility of the inscription and circumscription of a circle in and about a

quadrilateral figure ?

55. Point out those propositions in the Sixth Book in which Euclid's

definition of proportion is directly applied.

56. Explain briefly the advantages gained by the application of

analysis to the solution of Geometrical Problems.

57. In what cases are triangles proved to be equal in Euclid, and in

what cases are they proved to be similar ?

GEOMETEICAL EXEECISES ON BOOK YL

PROPOSITION I. PROBLEM.

To inscribe a square in a given triangle.

Analysis. Let ABC he the given triangle, of which the base BC^

and the perpendicular AD are given.

A

Let FGIIK be the required inscribed square.

Then BHG, BDA are similar triangles,

and G^^is to GB, as AD is to ABy

but G^i^is equal to GH\

therefore GFh to GB, as AD is to AB.

Let BF hQ joined and produced to meet a line drawn from A pa-

rallel to the base ^Cin the point E.

Then the triangles BGF, BAE are similar,

and ^^ is to AB, as GFis to GB,

but GFi& to GB, as AD is to AB ;

wherefore AE is to AB, as AD is to AB;

hence AE is equal to AD.

Synthesis. Through the vertex A, draw AE parallel to ^Cthe

base of the triangle,

make AE equal to AD,

join EB cutting AC in F,

through F, draw FG parallel to BC, and i^JT parallel to AD-,

also through G draw GH parallel to AD.

Then GHKF is the square required.

The different cases may be considered when the triangle is equi-

lateral, scalene, or isosceles, and when each side is taken as the base.

PROPOSITION II. THEOREM.

If from the extremities of any diameter of a given circle, perpendiculars

he drawn to any chord of the circle, they shall meet the chord, or the chord

produced in two points which are equidistant from the cetiter.

First, let the chord CD intersect the diameter AB in Z, but not

at right angles ; and from A, B, let AE, BFhe drawn perpendicular

to CD. Then the points F, E are equidistant from the center of the

chord CZ>.

Join EB, and from J the center of the circle, draw IG perpendi-

cular to CD, and produce it to meet EB in JÂ£,

GEOMETRICAL EXERCISES 303

A

Then IG bisects CD in G -, (ill. 2.)

and IG, AE being both perpendicular to CD, are parallel. (l. 29.)

Therefore BI\% to BH, as lA is to HE-, (VI. 2.)

and ^^is to FG, as HE is to GE-,

therefore J5Jis to FG, as I A is to GE-,

but BI is equal to I A :

therefore i^G' is equal to GE.

It is also manifest that DE is equal to CF.

When the chord does not intersect the diameter, the perpendicu-

lars intersect the chord produced.

PROPOSITION III. THEOREM.

If two diagonals of a regular pentagon he drawn to cut one another, the

greater segments will he equal to the side of the pentagon, and the diagonals

will cut one another in extreme and mean ratio. â–

Let the diagonals AC, BE be drawn from the extremities of the

side AB of the regular pentagon ABCDE, and intersect each other

in the point H.

Then BE and ^ C are cut in extreme and mean ratio in H, and

the greater segment of each is equal to the side of the pentagon.

Let the circle ABCDE be described about the pentagon, (iv. 14.)

Because EA, AB are equal to AB, BC, and they contain equal

angles ;

therefore the base EB is equal to the base A C, (l. 4.)

and the triangle EAB is equal to the triangle CBA,

and the remaining angles will be equal to the remaining angles,

each to each, to which the equal sides are opposite.

D

Therefore the angle BACh equal to the angle ABE-,

and the angle AHE is double of the angle BAH, (l. 32.)

but the angle EA C is also double of the angle BA C, (vi. 33.)

I therefore the angle HAE is equal to AHE,

and consequently HE is equal to EA, (i. 6.) or to AB.

And because BA is equal to AE,

the angle ABE is equal to the angle AEB j

304 GEOMETRICAL EXERCISES

but the angle ABE has been proved equal to BAH:

therefore the angle BEA is equal to the angle BAH:

and ABE is common to the two triangles ABE, ABH;

therefore the remaining angle BAE is equal to the remaining

angle AHB ;

and consequently the triangles ABE, ABH are equiangular ;

therefore EB is to BA, as AB to BH: but BA is equal to EH^

therefore EB is to EH, as EH is to BH,

but BE is greater than EH; therefore EH is greater than HB;

therefore BE has been cut in extreme and mean ratio in H.

Similarly, it may be shewn, that A C has also been cut in extreme

and mean ratio in H, and that the greater segment of it CH is equal

to the side of the pentagon.

PROPOSITION IV. PROBLEM.

Divide a given arc of a circle into two parts which shall have their chords

in a given ratio.

Analysis. Let A, Bhe the two given points in the circumference

of the circle, and Cthe point required to be found, such that when the

chords A C and BC are joined, tne lines A C and J5C shall have to one

another the ratio of E to F.

Draw CD touching the circle in C;

join AB and produce it to meet CD in D.

Since the angle BACis equal to the angle BCD, (ill. 32.)

and the angle CEB is common to the two triangles BBC, DAC;

therefore the third angle CBD in one, is equal to the third angle

DCA in the other, and the triangles are similar,

therefore AD is to DC, as i>C is to DB -, (vi. 4.)

hence also the square on AD is to the square on DC a,s AD is to

BD. (VI. 20. Cor.)

But AD is to AC, as DC is to CB, (vi. 4.)

and AD is to DC, as AC to CB, (v. 16.)

also the square on AD is to the square on DC, as the square on ^ C

is to the square on CB ;

but the square on AD is to the square on DC, as AD is to DB :

wherefore the square on ^ C is to the square on CB, as AD is to BD ;

but ^ C is to CB, as E is to F, (constr.)

therefore AD is to DB as the square on E is to the square on F.

Hence the ratio of AD to DB is given,

and AB is given in magnitude, because the points-^, B in the cir-

cumference of the circle are given.

ON BOOK VI.

805

Wherefore also the ratio of AD to AB is given, and also the mag-

nitude of AD.

Synthesis. Join AB and produce it to D, so that AD shall be to

BD, as the square on JE to the square on F.

From D draw DC to touch the circle in C, and join CB, CA.

Since AD is to DB, as the square on JEJ is to the square on F, (constr.)

and ^D is to DB, as the square on ^ C is to the square on J5C;

therefore the square on ^ C is to the square on BC, as the square on

F! is to the square on F,

and ACis to BC, as F is to F,

PROPOSITION V. PROBLEM.

A, B, C are given points. It is required to draio through any other point

in the same plane with A, B, and C, a straight line, such that the sum of its

distances from two of the given points, may be equal to its distance from the

third.

Analysis. Suppose F the point required, such that the line XFH

being drawn through any other point X, and AD, BE, CH perpen-

diculars on XFH, the sum of BE and CH is equal to AD.

I

^B Join AB, BC, CA, then ABCh a triangle.

^jDraw ^G^ to bisect the base J?Cin G, and draw GX perpendicular

to EF.

Then since BC is bisected in G,

the sum of the perpendiculars CH, BE is double of GK;

but CH and BE are equal to AD, (hyp.)

therefore AD must be double of GK;

but since AD is parallel to GK,

the triangles ADF, GKF are similar,

therefore AD h to AF, as GK is to GF;

but AD is double of GK, therefore ^i^is double of GF-,

and consequently, GFh one-third of ^G^ the line drawn from the

vertex of the triangle to the bisection of the base.

But ^ 6^ is a line given in magnitude and position,

therefore the point F is determined.

Synthesis. Join AB, A C, BC, and bisect the base BC o? the tri-

angle ABC in G; join AG and take G^i^ equal to one-third of GA ;

the line drawn through X and F will be the line required.

It is also obvious, that while the relative position of the points A,

C, remains the same, the point -F remains the same, wherever the

K

306 GEOMETRICAL EXERCISES

point X may be. The point X may therefore coincide with the point

jP, and when this is the case, the position of the line FX is left un-

determined. Hence the following porism.

A triangle being given in position, a point in it may be found,

such, that any straight line whatever being drawn through that point,

the perpendiculars drawn to this straight line from the two angles of

the triangle, which are on one side of it, will be together equal to the

perpendicular that is drawn to the same line from the angle on the

other side of it.

I.

6. Triangles and parallelograms of unequal altitudes are to each

other in the ratio compounded of the ratios of their bases and altitudes.

7. If ACB, ADB be two triangles upon the same base AB, and

between the same parallels, and if through the point in which two of

the sides (or two of the sides produced) intersect two straight lines be

drawn parallel to the other two sides so as to meet the base AB (or

AB produced) in points E and F. Prove that AE= BF.

8. In the base -4C of a triangle ^J?Ctake any point D; bisect

AD, DC,AB,BC, in F, F, G, R respectively : shew that EG is

equal to KF.

9. Construct an isosceles triangle equal to a given scalene triangle

and having an equal vertical angle with it.

10. If, in similar triangles, from any two equal angles to the

opposite sides, two straight lines be drawn making equal angles with

the homologous sides, these straight lines will have the same ratio as

the sides on which they fall, and will also divide those sides propor-

tionally.

11. Any three lines being drawn making equal angles with the

three sides of any triangle towards the same parts, and meeting one

another, will form a triangle similar to the original triangle.

12. BB, CD are perpendicular to the sides AB, AC of a triangle

ABC, and CE is drawn perpendicular to AD, meeting A B in E:

shew that the triangles ABC, ACE are similar.

13. In any triangle, if a perpendicular be let fall upon the base

from the vertical angle, the base will be to the sum of the sides, as the

difference of the sides to the difference or sum of the segments of the

base made by the perpendicular, according as it falls within or with-

out the triangle.

14. If triangles AEF, ABC have a common angle A, triangle

ABC: triangle AEFii AB.AC: AE.AF.

15. If one side of a triangle be produced, and the other shortened

by equal quantities, the line joining the points of section will be di-

vided by the base in the inverse ratio of the sides.

II.

16. Find two arithmetic means between two given straight lines.

17. To divide a given line in harmonical proportion.

ON BOOK VI. 307

18. To find, by a geometrical construction, an arithmetic,

geometric, and harmonic mean between two given lines.

19. Prove geometrically, that an arithmetic mean between two

quantities, is greater than a geometric mean. Also having given the

sum of two lines, and the excess of their arithmetic above their

geometric mean, find by a construction the lines themselves.

20. If through the point of bisection of the base of a triangle any

line be drawn, intersecting one side of the triangle, the other produced,

and a line drawn parallel to the base from the vertex, this line shall

be cut harmonically.

21. If a given straight line AJB be divided into any two parts in

the point C, it is required to produce it, so that the whole line

produced may be harmonically divided in C and JB.

22. If from a point without a circle there be drawn three straight

lines, two of which touch the circle, and the other cuts it, the line

which cuts the circle will be divided harmonically by the convex

circumference, and the chord which joins the points of contact.

III.

23. Shew geometrically that the diagonal and side of a square are

incommensurable.

24. If a straight line be divided in two given points, determine a

third point, such that its distances from the extremities, may be

proportional to its distances from the given points.

25. Determine two straight lines, such that the sum of their

squares may equal a given square, and their rectangle equal k given

rectangle.

26. Draw a straight line such that the perpendiculars let fall

from any point in it on two given lines may be in a given ratio.

27. If diverging lines cut a straight line, so that the whole is to

one extreme, as the other extreme is to the middle part, they will

intersect every other intercepted line in the same ratio.

28. It is required to cut ofi" a part of a given line so that the part

cut off may be a mean proportional between the remainder and

another given line.

29. It is required to divide a given finite straight line into two

parts, the squares of which shall have a given ratio to each other.

IV.

30. From the vertex of a triangle to the base, to draw a straight

line which shall be an arithmetic mean between the sides containing

the vertical angle.

31. From the obtuse angle of a triangle, it is required to draw a

line to the base, which shall be a mean proportional between the

segments of the base. How many answers does this question admit

of?

32. To draw a line from the vertex of a triangle to the base, which

shall be a mean proportional between the whole base and one segment,

33. If the perpendicular in a right-angled triangle divide the

hypotenuse in extreme and mean ratio, the less side is equal to the

alternate segment.

308 GEOMETRICAL EXERCISES

34. From the vertex of any triangle ABC, draw a straight line

meeting the base produced in D, so that the rectangle DB. DC- AIT.

35. To find a point P in the base J?C of a triangle produced, so

that BD being drawn parallel to A C, and meeting -4 jB produced to D,

AC: CB:: CB : BD.

36. If the triangle ^^C has the angle at C aright angle, and

from C a perpendicular be dropped on the opposite side intersecting

it in D, then AI> : DBiiAC: CB\

37. In any right-angled triangle, one side is to the other, as the

excess of the hypotenuse above the second, to the line cut off from the

first between the right angle and the line bisecting the opposite angle.

38. If on the two sides of a right-angled triangle squares be

described, the lines joining the acute angles of the triangle and the

opposite angles of the squares, will cut off equal segments from the

sides ; and each of these equal segments will be a mean proportional

between the remaining segments.

39. In any right-angled triangle ABC, (whose hj^otenuse is^^)

bisect the angle A by AD meeting CB in D, and prove that

2AC'''.AC''-Cn'::BC:CD.

40. On two given straight lines similar triangles are described.

Required to find a third, on which, if a triangle similar to them be

described, its area shall equal the difference of their areas.

41. In the triangle ABC, AC= 2.BC If CD, CE respectively

bisect the angle C, and the exterior angle formed by producing AC;

prove that the triangles CBD,ACD, ABC, CDE, have their areas as

1,2,3,4.

V.

42. It is required to bisect any triangle (1) by a line drawn parallel,

(2) by a line drawn perpendicular, to the base.

43. To divide a given triangle into two parts, having a given ratio

to one another, by a straight line di awn parallel to one of its sides.

44. Find three points in the sides of a triangle, such that, they

being joined, the triangle shall be divided into four equal triangles.

45. From a given point in the side of a triangle, to draw lines to

the sides which shall divide the triangle into any number of equal parts,

46. Any two triangles being given, to draw a straight line parallel

to a side of the greater, which shall cut off a triangle equal to the less.

VI.

47. The rectangle contained by two lines is a mean proportional

between their squares.

48. Describe a rectangular parallelogram which shall be equal to

a given square, and have its sides in a given ratio.

49. If from any two points within or without a parallelogram,

straight lines be drawn perpendicular to each of two adjacent sides

and intersecting each other, they form a parallelogram similar to the

former.

50. It is requii'ed to cut off from a rectangle a similar rectangle

which shall be any required part of it.

I

ON BOOK VI. 809

51. If from one angle ^ of a parallelogram a straight line be drawn

cutting the diagonal in E and the sides in P, Q, shew that

AE' = PE.EQ.

52. The diagonals of a trapezium, two of whose sides are parallel,

cut one another in the same ratio.

VII.

53. In a given circle place a straight line parallel to a given

straight line, and having a given ratio to it; the ratio not being

greater than that of the diameter to the given line in the circle.

54. In a given circle place a straight line, cutting two radii which

are jDerpendicular to each other, in such a manner, that the line itself

may be trisected.

55. AB is a diameter, and P any point in the circumference of a

circle ; AP and BP are joined and produced if necessary ; if from any

point C of AB, a perpendicular be drawn to AB meeting AP and BP

in points D and E respectively, and the circumference of the circle

in a point F, shew that CD is a third proportional of CE and CF.

56. If from the extremity of a diameter of a circle tangents be

drawn, any other tangent to the circle terminated by them is so

divided at its point of contact, that the radius of the circle is a mean

proportional between its segments.

57. From a given point without a circle, it is required to draw a

straight line to the concave circumference, which shall be divided in a

given ratio at the point where it intersects the convex circumference.

58. From what point in a circle must a tangent be drawn, so that

a perpendicular on it from a given point in the circumference may be

cut by the circle in a given ratio ?

59. Through a given point within a given circle, to draw a

straight line such that the parts of it intercepted between that point

and the circumference, may have a given ratio.

60. Let the two diameters AB, CD, of the circle AD B Che at

right angles to each other, draw any chord EF, join CE, CF, meeting

AB in (rand^; prove that the triangles CG^7/and C-2J-F are similar.

61. A circle, a straight line, and a point being given in position,

required a point in the line, such that a line drawn from it to the

given point may be equal to a line drawn from it touching the circle.

^Yhat must be the relation among the data, that the problem may

become porismatic, i.e. admit of innumerable solutions ?

VIII.

62. Prove that there may be two, but not more than two, similar

triangles in the same segment of a circle.

63. If as in Euclid vi. 3, the vertical angle BA C of the triangle

BAC be bisected by AD, and BA be produced to meet CE drawn

parallel to AD in E; shew that AD will be a tangent to the circle

described about the triangle EAC.

64. If a triangle be inscribed in a circle, and from its vertex, lines

be drawn ])arallel to the tangents at the extremities of its base, they

15. Point out clearly the difference in the proofs of the two latter cases

in Euc. VI. 7.

16. From the corollary of Euc. vi. 8, deduce a proof of Euc. i. 47.

17. Shew how the last two properties stated in Euc. vi. 8. Cor. may

be deduced from Euc. i. 47 ; ii. 2 ; vi. 17.

18. Given the nth part of a straight line, find by a Geometrical con-

struction, the (n + l)tli part.

19. Define what is meant by a mean proportional between two given

lines : and find a mean proportional between the lines whose lengths are

4 and 9 units resj)ectively. Is the method you employ suggested by any

Propositions in any of the first four books ?

20. Determine a third proportional to two lines of 5 and 7 units : and

a fourth proportional to three lines of o, 7, 9, units.

21. Find a straight line which shall have to a given straight line, the

ratio of 1 to Vo.

22. Define reciprocal figures. Enunciate the propositions proved re-

specting such figures in the Sixth Book.

23. Give the corollary, Euc. vi. 8, and prove thence that the Arith-

metic mean is greater than the Geometric between the same extremes.

24. If two equal triangles have two angles together equal to two

right angles, the sides about those angles are reciprocally proportional.

25. Give Algebraical proofs of Prop. 16 and 17 of Book vi.

26. Enunciate and prove the converse of Euc. vi. 15.

27. Explain what is meant by saying, that "similar triangles are in the

duplicate ratio of their homologous sides."

28. What are the data which determine triangles both in species and

magnitude ? How are those data expressed in Geometry ?

29. If the ratio of the homologous sides of two triangles be as 1 to

4, what is the ratio of the triangles ? And if the ratio of the triangles be

as 1 to 4, what is the ratio of the homologous sides ?

30. Shew that one of the triangles in the figure, Euc. iv. 10, is a mean

proportional between the other two.

31. What is the algebraical interpretation of Euc. vi. 19 ?

32. From j'-our definition of Proportion, prove that the diagonals of

a square are in the same proportion as their sides.

33. What propositions does Euclid prove respecting similar polygons ?

34. The parallelograms about the diameter of a parallelogram are similar

to the whole and to one another. Shew when they are equal. â–

35. Prove Algebraically, that the areas (1) of similar triangles and (2)

of similar parallelograms are proportional to the squares of their homo-

logous sides.

36. How is it shewn that equiangular parallelograms have to one

another the ratio which is compounded of the ratios of their bases and al-

titudes ?

37. To find two lines which shall have to each other, the ratio com-

pounded of the ratios of the lines A to B, and C to D.

38. State the force of the condition ** similarly described ;" and shew

that, on a given straight line, there may be described as many polygons

of different magnitudes, similar to a given polygon, as there are sides of

Afferent lengths in the polygon.

â€¢

QUESTIONS ON BOOK VF. oOl

39. Describe a triangle similar to a given triangle, and having its

area double that of the given triangle.

40. The three sides of a triangle are 7, 8, 9 units respectively; deter-

mine the length of the lines which meeting the base, and the base produced,

bisect the interior angle opposite to the greatest side of the triangle,

and the adjacent exterior angle.

41. The three sides of a triangle are 3, 4, 5 inches respectively ; find

the lengths of the external segments of the sides determined by the lines

which bisect the exterior angles of the triangle.

42. What are the segments into which the hypotenuse of a right-

angled triangle is divided by a perpendicular drawn from the right angle,

if the sides containing it are a and 3a units respectively ?

43. If the three sides of a triangle be 3, 4, 5 units respectively : what

are the parts into which they are divided by the lines which bisect the

angles opposite to them ?

44. If the homologous sides of two triangles be as 3 to 4, and the area

of one triangle be known to contain 100 square units ; how many square

units are contained in the area of the other triangle ?

45. Prove that if BD be taken in AB produced (fig. Euc. vi. 30)

equal to the greater segment AC^ then AD is divided in extreme and

mean ratio in the point B.

Shew also, that in the series 1, 1, 2, 3, 5, 8, &c. in which each term is

the sum of the two preceding terms, the last two terms perpetually ap-

proach to the proportion of the segments of a line divided in extreme and

mean ratio. Find a general expression (free from surds) for the Mth term

of this series.

46. The parts of a line divided in extreme and mean ratio are incom-

mensurable with each other.

47. Shew that in Euclid's figure (Euc. ii. 11.) four other lines, besides

the given line, are divided in the required manner.

48. Enunciate Euc. vi. 31. What theorem of a previous book is in-

cluded in this proposition ?

49. What is the superior limit, as to magnitude, of the angle at the

circumference in Euc. vi. 33 ? Shew that the proof may be extended by

withdrawing the usually supposed restriction as to angular magnitude ;

and then deduce, as a corollary, the proposition respecting the magnitudes

of angles in segments greater than, equal to, or less than a semicircle.

50. The sides of a triangle inscribed in a circle are a, 6, c, units respec-

tively : find by Euc. vi. c, the radius of the circumscribing circle.

51. Enunciate the converse of Euc. vi. d.

52. Shew independently that Euc. vi. d, is true when the quadri-

lateral figure is rectangular.

53. Shew that the rectangles contained by the opposite sides of a

quadrilateral figure which does not admit of having a circle described

about it, are together greater than the rectangle contained by the diagonals.

54. What diff'erent conditions may be stated as essential to the possi-

bility of the inscription and circumscription of a circle in and about a

quadrilateral figure ?

55. Point out those propositions in the Sixth Book in which Euclid's

definition of proportion is directly applied.

56. Explain briefly the advantages gained by the application of

analysis to the solution of Geometrical Problems.

57. In what cases are triangles proved to be equal in Euclid, and in

what cases are they proved to be similar ?

GEOMETEICAL EXEECISES ON BOOK YL

PROPOSITION I. PROBLEM.

To inscribe a square in a given triangle.

Analysis. Let ABC he the given triangle, of which the base BC^

and the perpendicular AD are given.

A

Let FGIIK be the required inscribed square.

Then BHG, BDA are similar triangles,

and G^^is to GB, as AD is to ABy

but G^i^is equal to GH\

therefore GFh to GB, as AD is to AB.

Let BF hQ joined and produced to meet a line drawn from A pa-

rallel to the base ^Cin the point E.

Then the triangles BGF, BAE are similar,

and ^^ is to AB, as GFis to GB,

but GFi& to GB, as AD is to AB ;

wherefore AE is to AB, as AD is to AB;

hence AE is equal to AD.

Synthesis. Through the vertex A, draw AE parallel to ^Cthe

base of the triangle,

make AE equal to AD,

join EB cutting AC in F,

through F, draw FG parallel to BC, and i^JT parallel to AD-,

also through G draw GH parallel to AD.

Then GHKF is the square required.

The different cases may be considered when the triangle is equi-

lateral, scalene, or isosceles, and when each side is taken as the base.

PROPOSITION II. THEOREM.

If from the extremities of any diameter of a given circle, perpendiculars

he drawn to any chord of the circle, they shall meet the chord, or the chord

produced in two points which are equidistant from the cetiter.

First, let the chord CD intersect the diameter AB in Z, but not

at right angles ; and from A, B, let AE, BFhe drawn perpendicular

to CD. Then the points F, E are equidistant from the center of the

chord CZ>.

Join EB, and from J the center of the circle, draw IG perpendi-

cular to CD, and produce it to meet EB in JÂ£,

GEOMETRICAL EXERCISES 303

A

Then IG bisects CD in G -, (ill. 2.)

and IG, AE being both perpendicular to CD, are parallel. (l. 29.)

Therefore BI\% to BH, as lA is to HE-, (VI. 2.)

and ^^is to FG, as HE is to GE-,

therefore J5Jis to FG, as I A is to GE-,

but BI is equal to I A :

therefore i^G' is equal to GE.

It is also manifest that DE is equal to CF.

When the chord does not intersect the diameter, the perpendicu-

lars intersect the chord produced.

PROPOSITION III. THEOREM.

If two diagonals of a regular pentagon he drawn to cut one another, the

greater segments will he equal to the side of the pentagon, and the diagonals

will cut one another in extreme and mean ratio. â–

Let the diagonals AC, BE be drawn from the extremities of the

side AB of the regular pentagon ABCDE, and intersect each other

in the point H.

Then BE and ^ C are cut in extreme and mean ratio in H, and

the greater segment of each is equal to the side of the pentagon.

Let the circle ABCDE be described about the pentagon, (iv. 14.)

Because EA, AB are equal to AB, BC, and they contain equal

angles ;

therefore the base EB is equal to the base A C, (l. 4.)

and the triangle EAB is equal to the triangle CBA,

and the remaining angles will be equal to the remaining angles,

each to each, to which the equal sides are opposite.

D

Therefore the angle BACh equal to the angle ABE-,

and the angle AHE is double of the angle BAH, (l. 32.)

but the angle EA C is also double of the angle BA C, (vi. 33.)

I therefore the angle HAE is equal to AHE,

and consequently HE is equal to EA, (i. 6.) or to AB.

And because BA is equal to AE,

the angle ABE is equal to the angle AEB j

304 GEOMETRICAL EXERCISES

but the angle ABE has been proved equal to BAH:

therefore the angle BEA is equal to the angle BAH:

and ABE is common to the two triangles ABE, ABH;

therefore the remaining angle BAE is equal to the remaining

angle AHB ;

and consequently the triangles ABE, ABH are equiangular ;

therefore EB is to BA, as AB to BH: but BA is equal to EH^

therefore EB is to EH, as EH is to BH,

but BE is greater than EH; therefore EH is greater than HB;

therefore BE has been cut in extreme and mean ratio in H.

Similarly, it may be shewn, that A C has also been cut in extreme

and mean ratio in H, and that the greater segment of it CH is equal

to the side of the pentagon.

PROPOSITION IV. PROBLEM.

Divide a given arc of a circle into two parts which shall have their chords

in a given ratio.

Analysis. Let A, Bhe the two given points in the circumference

of the circle, and Cthe point required to be found, such that when the

chords A C and BC are joined, tne lines A C and J5C shall have to one

another the ratio of E to F.

Draw CD touching the circle in C;

join AB and produce it to meet CD in D.

Since the angle BACis equal to the angle BCD, (ill. 32.)

and the angle CEB is common to the two triangles BBC, DAC;

therefore the third angle CBD in one, is equal to the third angle

DCA in the other, and the triangles are similar,

therefore AD is to DC, as i>C is to DB -, (vi. 4.)

hence also the square on AD is to the square on DC a,s AD is to

BD. (VI. 20. Cor.)

But AD is to AC, as DC is to CB, (vi. 4.)

and AD is to DC, as AC to CB, (v. 16.)

also the square on AD is to the square on DC, as the square on ^ C

is to the square on CB ;

but the square on AD is to the square on DC, as AD is to DB :

wherefore the square on ^ C is to the square on CB, as AD is to BD ;

but ^ C is to CB, as E is to F, (constr.)

therefore AD is to DB as the square on E is to the square on F.

Hence the ratio of AD to DB is given,

and AB is given in magnitude, because the points-^, B in the cir-

cumference of the circle are given.

ON BOOK VI.

805

Wherefore also the ratio of AD to AB is given, and also the mag-

nitude of AD.

Synthesis. Join AB and produce it to D, so that AD shall be to

BD, as the square on JE to the square on F.

From D draw DC to touch the circle in C, and join CB, CA.

Since AD is to DB, as the square on JEJ is to the square on F, (constr.)

and ^D is to DB, as the square on ^ C is to the square on J5C;

therefore the square on ^ C is to the square on BC, as the square on

F! is to the square on F,

and ACis to BC, as F is to F,

PROPOSITION V. PROBLEM.

A, B, C are given points. It is required to draio through any other point

in the same plane with A, B, and C, a straight line, such that the sum of its

distances from two of the given points, may be equal to its distance from the

third.

Analysis. Suppose F the point required, such that the line XFH

being drawn through any other point X, and AD, BE, CH perpen-

diculars on XFH, the sum of BE and CH is equal to AD.

I

^B Join AB, BC, CA, then ABCh a triangle.

^jDraw ^G^ to bisect the base J?Cin G, and draw GX perpendicular

to EF.

Then since BC is bisected in G,

the sum of the perpendiculars CH, BE is double of GK;

but CH and BE are equal to AD, (hyp.)

therefore AD must be double of GK;

but since AD is parallel to GK,

the triangles ADF, GKF are similar,

therefore AD h to AF, as GK is to GF;

but AD is double of GK, therefore ^i^is double of GF-,

and consequently, GFh one-third of ^G^ the line drawn from the

vertex of the triangle to the bisection of the base.

But ^ 6^ is a line given in magnitude and position,

therefore the point F is determined.

Synthesis. Join AB, A C, BC, and bisect the base BC o? the tri-

angle ABC in G; join AG and take G^i^ equal to one-third of GA ;

the line drawn through X and F will be the line required.

It is also obvious, that while the relative position of the points A,

C, remains the same, the point -F remains the same, wherever the

K

306 GEOMETRICAL EXERCISES

point X may be. The point X may therefore coincide with the point

jP, and when this is the case, the position of the line FX is left un-

determined. Hence the following porism.

A triangle being given in position, a point in it may be found,

such, that any straight line whatever being drawn through that point,

the perpendiculars drawn to this straight line from the two angles of

the triangle, which are on one side of it, will be together equal to the

perpendicular that is drawn to the same line from the angle on the

other side of it.

I.

6. Triangles and parallelograms of unequal altitudes are to each

other in the ratio compounded of the ratios of their bases and altitudes.

7. If ACB, ADB be two triangles upon the same base AB, and

between the same parallels, and if through the point in which two of

the sides (or two of the sides produced) intersect two straight lines be

drawn parallel to the other two sides so as to meet the base AB (or

AB produced) in points E and F. Prove that AE= BF.

8. In the base -4C of a triangle ^J?Ctake any point D; bisect

AD, DC,AB,BC, in F, F, G, R respectively : shew that EG is

equal to KF.

9. Construct an isosceles triangle equal to a given scalene triangle

and having an equal vertical angle with it.

10. If, in similar triangles, from any two equal angles to the

opposite sides, two straight lines be drawn making equal angles with

the homologous sides, these straight lines will have the same ratio as

the sides on which they fall, and will also divide those sides propor-

tionally.

11. Any three lines being drawn making equal angles with the

three sides of any triangle towards the same parts, and meeting one

another, will form a triangle similar to the original triangle.

12. BB, CD are perpendicular to the sides AB, AC of a triangle

ABC, and CE is drawn perpendicular to AD, meeting A B in E:

shew that the triangles ABC, ACE are similar.

13. In any triangle, if a perpendicular be let fall upon the base

from the vertical angle, the base will be to the sum of the sides, as the

difference of the sides to the difference or sum of the segments of the

base made by the perpendicular, according as it falls within or with-

out the triangle.

14. If triangles AEF, ABC have a common angle A, triangle

ABC: triangle AEFii AB.AC: AE.AF.

15. If one side of a triangle be produced, and the other shortened

by equal quantities, the line joining the points of section will be di-

vided by the base in the inverse ratio of the sides.

II.

16. Find two arithmetic means between two given straight lines.

17. To divide a given line in harmonical proportion.

ON BOOK VI. 307

18. To find, by a geometrical construction, an arithmetic,

geometric, and harmonic mean between two given lines.

19. Prove geometrically, that an arithmetic mean between two

quantities, is greater than a geometric mean. Also having given the

sum of two lines, and the excess of their arithmetic above their

geometric mean, find by a construction the lines themselves.

20. If through the point of bisection of the base of a triangle any

line be drawn, intersecting one side of the triangle, the other produced,

and a line drawn parallel to the base from the vertex, this line shall

be cut harmonically.

21. If a given straight line AJB be divided into any two parts in

the point C, it is required to produce it, so that the whole line

produced may be harmonically divided in C and JB.

22. If from a point without a circle there be drawn three straight

lines, two of which touch the circle, and the other cuts it, the line

which cuts the circle will be divided harmonically by the convex

circumference, and the chord which joins the points of contact.

III.

23. Shew geometrically that the diagonal and side of a square are

incommensurable.

24. If a straight line be divided in two given points, determine a

third point, such that its distances from the extremities, may be

proportional to its distances from the given points.

25. Determine two straight lines, such that the sum of their

squares may equal a given square, and their rectangle equal k given

rectangle.

26. Draw a straight line such that the perpendiculars let fall

from any point in it on two given lines may be in a given ratio.

27. If diverging lines cut a straight line, so that the whole is to

one extreme, as the other extreme is to the middle part, they will

intersect every other intercepted line in the same ratio.

28. It is required to cut ofi" a part of a given line so that the part

cut off may be a mean proportional between the remainder and

another given line.

29. It is required to divide a given finite straight line into two

parts, the squares of which shall have a given ratio to each other.

IV.

30. From the vertex of a triangle to the base, to draw a straight

line which shall be an arithmetic mean between the sides containing

the vertical angle.

31. From the obtuse angle of a triangle, it is required to draw a

line to the base, which shall be a mean proportional between the

segments of the base. How many answers does this question admit

of?

32. To draw a line from the vertex of a triangle to the base, which

shall be a mean proportional between the whole base and one segment,

33. If the perpendicular in a right-angled triangle divide the

hypotenuse in extreme and mean ratio, the less side is equal to the

alternate segment.

308 GEOMETRICAL EXERCISES

34. From the vertex of any triangle ABC, draw a straight line

meeting the base produced in D, so that the rectangle DB. DC- AIT.

35. To find a point P in the base J?C of a triangle produced, so

that BD being drawn parallel to A C, and meeting -4 jB produced to D,

AC: CB:: CB : BD.

36. If the triangle ^^C has the angle at C aright angle, and

from C a perpendicular be dropped on the opposite side intersecting

it in D, then AI> : DBiiAC: CB\

37. In any right-angled triangle, one side is to the other, as the

excess of the hypotenuse above the second, to the line cut off from the

first between the right angle and the line bisecting the opposite angle.

38. If on the two sides of a right-angled triangle squares be

described, the lines joining the acute angles of the triangle and the

opposite angles of the squares, will cut off equal segments from the

sides ; and each of these equal segments will be a mean proportional

between the remaining segments.

39. In any right-angled triangle ABC, (whose hj^otenuse is^^)

bisect the angle A by AD meeting CB in D, and prove that

2AC'''.AC''-Cn'::BC:CD.

40. On two given straight lines similar triangles are described.

Required to find a third, on which, if a triangle similar to them be

described, its area shall equal the difference of their areas.

41. In the triangle ABC, AC= 2.BC If CD, CE respectively

bisect the angle C, and the exterior angle formed by producing AC;

prove that the triangles CBD,ACD, ABC, CDE, have their areas as

1,2,3,4.

V.

42. It is required to bisect any triangle (1) by a line drawn parallel,

(2) by a line drawn perpendicular, to the base.

43. To divide a given triangle into two parts, having a given ratio

to one another, by a straight line di awn parallel to one of its sides.

44. Find three points in the sides of a triangle, such that, they

being joined, the triangle shall be divided into four equal triangles.

45. From a given point in the side of a triangle, to draw lines to

the sides which shall divide the triangle into any number of equal parts,

46. Any two triangles being given, to draw a straight line parallel

to a side of the greater, which shall cut off a triangle equal to the less.

VI.

47. The rectangle contained by two lines is a mean proportional

between their squares.

48. Describe a rectangular parallelogram which shall be equal to

a given square, and have its sides in a given ratio.

49. If from any two points within or without a parallelogram,

straight lines be drawn perpendicular to each of two adjacent sides

and intersecting each other, they form a parallelogram similar to the

former.

50. It is requii'ed to cut off from a rectangle a similar rectangle

which shall be any required part of it.

I

ON BOOK VI. 809

51. If from one angle ^ of a parallelogram a straight line be drawn

cutting the diagonal in E and the sides in P, Q, shew that

AE' = PE.EQ.

52. The diagonals of a trapezium, two of whose sides are parallel,

cut one another in the same ratio.

VII.

53. In a given circle place a straight line parallel to a given

straight line, and having a given ratio to it; the ratio not being

greater than that of the diameter to the given line in the circle.

54. In a given circle place a straight line, cutting two radii which

are jDerpendicular to each other, in such a manner, that the line itself

may be trisected.

55. AB is a diameter, and P any point in the circumference of a

circle ; AP and BP are joined and produced if necessary ; if from any

point C of AB, a perpendicular be drawn to AB meeting AP and BP

in points D and E respectively, and the circumference of the circle

in a point F, shew that CD is a third proportional of CE and CF.

56. If from the extremity of a diameter of a circle tangents be

drawn, any other tangent to the circle terminated by them is so

divided at its point of contact, that the radius of the circle is a mean

proportional between its segments.

57. From a given point without a circle, it is required to draw a

straight line to the concave circumference, which shall be divided in a

given ratio at the point where it intersects the convex circumference.

58. From what point in a circle must a tangent be drawn, so that

a perpendicular on it from a given point in the circumference may be

cut by the circle in a given ratio ?

59. Through a given point within a given circle, to draw a

straight line such that the parts of it intercepted between that point

and the circumference, may have a given ratio.

60. Let the two diameters AB, CD, of the circle AD B Che at

right angles to each other, draw any chord EF, join CE, CF, meeting

AB in (rand^; prove that the triangles CG^7/and C-2J-F are similar.

61. A circle, a straight line, and a point being given in position,

required a point in the line, such that a line drawn from it to the

given point may be equal to a line drawn from it touching the circle.

^Yhat must be the relation among the data, that the problem may

become porismatic, i.e. admit of innumerable solutions ?

VIII.

62. Prove that there may be two, but not more than two, similar

triangles in the same segment of a circle.

63. If as in Euclid vi. 3, the vertical angle BA C of the triangle

BAC be bisected by AD, and BA be produced to meet CE drawn

parallel to AD in E; shew that AD will be a tangent to the circle

described about the triangle EAC.

64. If a triangle be inscribed in a circle, and from its vertex, lines

be drawn ])arallel to the tangents at the extremities of its base, they

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