Euclid. # A text-book of Euclid's Elements : for the use of schools : Books I-VI and XI online

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.-. the z EDO = the z FDB = the z A.

In like manner it may be proved that

the z DEC = the z FEA=:the z B,

and the Z DFB = the z EFA = the z C.

Corollary, (ii) The triangles DEC, AEF, DBF are equiangular

to one another and to the triangle ABC.

Note. If the angle BAC is obtuse, then the perpendiculars BE, CF

bisect externally the corresponding angles of the pedal triangle.

Ji. E. 15

226 EUCIill/.S ELEMENTS.

21. In a III/ triinifjlr, if the perpendiculars drawn from the vertices

on the opposite s/J, s art' produced to meet the circumscribed circle,

then each side bist'ctn that portion of the line perpendicular to it which

lies between the orthocentre and the circumference.

Let ABC be a triangle in which the perpen-

diculars AD, BE are drawn, intersecting at O the

orthocentre; and let AD be produced to meet

the o*Â® of the circumscribing circle at G :

then shall DO=DG.

Join BG.

Then in the two a" OEA, ODB,

the z OEA = the z ODB, being rt. angles;

and the Z EOA = the vert. opp. Z DOB;

.â€¢. the remaining Z EAO = the remaining Z DBO. i. 32.

But the z CAG=the z CBG, in the same segment;

.-. the Z DBO = the z DBG.

Then in the a" DBO, DBG,

(the Z DBO = the Z DBG, Proved.

Because jthe z BDO = the z BDG,

( and BD is common;

.-. DO = DG. 1.26.

Q. E. D.

22. In an acute-angled triangle the three sides are the external

bisectors of the angles of the pedal triangle : and in an obtuse-angled

triangle the sides containing the obtuse angle are the internal bisectors

of the corresponding angles of the pedal triangle.

23. If O is the orthocentre of the triangle ABC, sheic that the

angles BOC, BAC are supplementary.

24. If O is the orthocentre of the triangle ABC, then any one of

the four points O, A, B, C is the orthocentre of the triangle whose

vertices are the other three.

25. The three circles lohich pass through two vertices of a triangU

and its orthocentre are each equal to the circle circumscribed about the

triangle.

26. D , E are taken on the circumference of a semicircle described

on a given straight line AB : the chords AD, BE and AE, BD

intersect (produced if necessary) at F and G : shew that FG is per-

pendicular to AB.

27. A BCD is a parallelogram; AE and CE are drawn at right

angles to AB, and CB respectively: shew that ED, if produced, will

be perjiendicular to AC.

THEOREMS AND EXAMPLES ON BOOK III. 227

28. ABC is a triangle, O is its orthocentre, and AK a diameter

of the circumscribed circle: shew that BOCK is a parallelogram.

29. The orthocentre of a triangle is joined to the middle point of

the base, and the joining line is produced to meet the circumscribed

circle : prove that it will meet it at the same point as the diameter

which passes through the vertex.

30. The perpendicular from the vertex of a triangle on the base,

and the straight line joining the orthocentre to the middle point of

the base, are produced to meet the circumscribed circle at P and Gt :

shew that PQl is parallel to the base.

31. The distance of each vertex of a triangle from the orthocentre

is double of the perpendicular drawn from the centre of the circum-

scribed circle on the opposite side.

32. Three circles are described each passing through the ortho-

centre of a triangle and two of its vertices: shew that the triangle

formed by joining their centres is equal in all respects to the original

triangle.

33. ABC is a triangle inscribed in a circle, and the bisectors of its

angles which intersect at O are produced to meet the circumference in

PQR : shew that O is the orthocentre of the triangle PQR.

34. Construct a triangle, having given a vertex, the orthocentre,

and the centre of the circumscribed circle.

Loci.

35. Given the base and vertical angle of a triangle, find the locus

of its orthocentre.

Let BC be the given base, and X the

^iven angle ; and let BAC be any triangle

on the base BC, having its vertical z A

equal to the Z X.

Draw the perpÂ« BE, CF, intersecting

at the orthocentre O.

It is required to find the locus of O.

Since the z ' OFA, OEA axe rt. angles,

.-. the points O, F, A, E are concyclic ;

.â€¢.the Z FOE is the supplement of the z A:

.-. the vert. opp. z BOC is the supplement of the Z A.

But the Z A is constant, being always equal to the Z X ;

.â€¢. its supplement is constant ;

that is, the A BOC has a fixed base, and constant vertical angle;

hence the locus of its vertex O is the arc of a segment of which BC is

the chord. [See p. 187.]

15-2

228

KUCLID8 ELEMENTS.

36. Given the base and vertical angle of a triangle, find the luciis

of the intersection of the bisectors of its angles.

Let BAG be any triangle on tlie given

base BC, liaviug its vertical angle equal to

the given z X; and let Al, Bl, CI be the

bisectors of its angles: [see Ex. 2, p. 103.]

it is required to find the locus of the

point I.

Denote the angles of the A ABC by

A, B,C; and let the z BIC be denoted by I.

Then from the a BIC,

(i) l + JB + *C = twort.

and from the a ABC,

A + B + C = two rt. angles ;

(ii) so that ^A + ^B + ^C = one rt. angle,

.". , taking the differences of the equals in (i) and (ii),

I - ^ A == one rt. angle :

l=one rt. angle + ^A.

or,

X;

I. 32.

But A is constant, being always equal to the z

.-. I is constant :

.". , since the base BC is fixed, the locus of I is the arc of a segment

of which BC is the chord.

37. Given the base and vertical angle of a triangle, find the locus

of the centroid, that is, the intersection of the medians.

Let BAC be any triangle on the given

base BC, having its vertical angle equal

to the given angle S; let the medians AX,

BY, CZ intersect at the centroid G [see

Ex. 4, p. 105] :

it is required to find the locus of the point G .

Through G draw GP, GQ par' to AB

and AC respectively.

Then ZG is a third part of ZC;

Ex. 4, p. 105.

and since GP is par' to ZB,

.-. BP is a third part of BC.

Similarly QC is a third part of BC ;

.â€¢. P and Q are fixed points.

Now since PG, GQ are par' respectively to BA, AC,

.-. the z PGQ = the z BAC,

= the z S,

that is, the Z PGQ is constant;

and since the base PQ is fixed,

.-. the locus of G is the arc of a segment of which PQ is the chord.

Ex. 19, p. 09.

Constr.

' I. 29.

THKOIIKMS AND EXAMPLES ON IJOOK III. 229

Ohs. In this problem the points A and G move on the arcs of

similar segments.

38. Given the base and the vertical angle of a triangle ; find the

locus of the intersection of the bisectors of the exterior base angles.

39. Through the extremities of a given straight line AB any two

parallel straight lines AP, BQ are drawn ; find the lociis of the inter-

section of the bisectors of the angles PAB, QBA.

40. Find the locus of the middle points of chords of a circle drawn

through a fixed point.

Distinguish between the cases when the given point is within,

on, or without the circumference.

41. Find the locus of the points of contact of tangents drawn

from a fixed point to a system of concentric circles.

42. Find the locus of the intersection of straight lines which pass

through two fixed points on a circle and intercept on its circumference

an arc of constant length.

43. A and B are two fixed points on the circumference of a circle,

and PQ is any diameter : find the locus of the intersection of PA and

QB.

44. BAG is any triangle described on the fixed base BG and

having a constant vertical angle ; and BA is produced to P, so that

BP is equal to the sum of the sides containing the vertical angle: find

the locus of P.

45. AB is a fixed chord of a circle, and AC is a moveable chord

passing through A: if the parallelogram GB is completed, find the

locus of the intersection of its diagonals.

46. A straight rod PQ slides between two rulers placed at right

angles to one another, and from its extremities PX, QX are drawn

perpendicular to the rulers: find the locus of X.

47. Two circles whose centres are G and D, intersect at A and B :

through A, any straight line PAQ is drawn terminated by the circum-

ferences ; and PG, QD intersect at X: find the locus of X, and shew

that it passes through B. [Ex. 9, p. 216.]

48. Two circles intersect at A and B, and through P, any point

on the circumference of one of them, two straight lines PA, PB

are drawn, and produced if necessary, to cut the other circle at X

and Y: find the locus of the intersection of AY and BX.

49. Two circles intersect at A and B; HAK is a fixed straight

line drawn through A and terminated by the circumferences, and

PAQ is any other straight line similarly drawn: find the locus of the

intersection of HP and QK. [Ex. 3, p. 186.]

::;3() KUtJ.IDS KLKMKNTS.

50. Two segments of circles are on the same chord AB and on

the same side of it ; and P and Q are any points one on each arc :

lintl the locus of the intersection of the bisectors of the angles PAQ,

PBQ.

51. Two circles intersect at A and B ; and through A any straight

line PAQ is drawn terminated by the circumferences : find the locus of

the middle point of PQ.

Miscellaneous Examples on Angles in a Circle.

52. ABC is a triangle, and circles are drawn through B, C, cutting

the sides in P, Q, P', Q', ... : shew that PQ, P'Q' ... are parallel to one

another and to tlie tangent drawn at A to the circle circumscribed

about the triangle.

53. Two circles intersect at B and C, and from any point A, on

the circumference of one of them, AB, AC are drawn, and produced if

necessary, to meet the other at D and E : shew that D E is parallel to

the tangent at A.

64. A secant PAB and a tangent PT are drawn to a circle from

an external point P; and tbe bisector of the angle ATB meets AB at

C : shew that PC is equal to PT.

55. From a point A on the circumference of a circle two chords

AB, AC are drawn, and also the diameter AF: if AB, AC are produced

to meet the tangent at F in D and E, shew that the triangles ABC,

AED are equiangular to one another.

56. O is any point within a triangle ABC, and CD, OE, OF are

drawn j)erpendicular to BC, CA, AB respectively : shew that the

angle BOC is equal to the sum of the angles BAC, EDF.

57. If two tangents are drawn to a circle from an external point,

shew that they contain an angle equal to the difference of the angles

in the segments cut off by the chord of contact.

58. Two circles intersect, and through a point of section a straight

line is drawn bisecting the angle between the diameters through that

point : shew that this straight line cuts off similar segments from the

two circles.

59. Two equal circles intersect at A and B ; and from centre

A, with any radius less than AB a third circle is described cutting the

given circles on the same side of A B at C and D: shew that the points

B, C, D are collinear.

60. ABC and A'B'C are two triangles inscribed in a circle, so that

AB, AC are respectively parallel to A'B', A'C : shew that BC is

parallel to B'C,

THEOREMS AND EXAMPLES ON BOOK III. 231

01. Two circles intersect at A and B, and through A two straight

lines HAK, PAQ are drawn terminated by the circumferences : if

HP and KQ intersect at X, shew that the points H, B, K, X are

concyclic.

62. Describe a circle touching a given straight line at a given

point, so that tangents drawn to it from two fixed points in the given

line may be parallel. [See Ex. 10, p. 183.]

G3. C is the centre of a circle, and CA, CB two fixed radii: if

from any point P on the arc AB perpendiculars PX, PY are drawn to

CA and CB, shew that the distance XY is constant.

64. AB is a chord of a circle, and P any point in its circum-

ference ; PM is drawn perpendicular to AB, and AN is drawn perpen-

dicular to the tangent at P : shew that MN is parallel to PB.

65. P is any point on the circumference of a circle of which AB is

a fixed diameter, and PN is drawn perpendicular to AB ; on AN and

BN as diameters circles are described, which are cut by AP, BP

at X and Y : shew that XY is a common tangent to these circles;

66. Upon the same chord and on the same side of it three seg-

ments of circles are described containing respectively a given angle,

its supplement and a right angle: shew that the intercept made by the

two former segments upon any straight line drawn through an ex-

tremity of the given chord is bisected by the latter segment.

67. Two straight lines of indefinite length touch a given circle,

and any chord is drawn so as to be bisected by the chord of contact :

if the former chord is produced, shew that the intercepts between the

circumference and the tangents are equal.

68. Two circles intersect one another: through one of the points

of contact draw a straight line of given length terminated by the cir-

cumferences.

69. On the three sides of any triangle equilateral triangles are

described remote from the given triangle : shew that the circles de-

scribed about them intersect at a point.

70. On BC, CA, AB the sides of a triangle ABC, any points

P, Q, R are taken; shew that the circles described about the triangles

AQR, BRP, CPQ meet in a point.

71. Find a point within a triangle at which the sides subtend

equal angles.

72. Describe an equilateral triangle so that its sides may pass

through three given points.

73. Describe a triangle equal in all respects to a given triangle,

and having its sides passing through three given points.

â€¢2:i-2 KUCLID'b KLKMENTS.

Simbon'b Line.

74. If from any point on the circumference of the circle circum-

scribed about a triangle, pei-pendiculars are drawn to the three siden, the

feet of these perpendiculars are coUinear.

Let P be any point on the o"" of the

circle circumscribed about the a ABC ;

and let PD, PE, PF be the perp" drawn

from P to the three sides.

It is required to prove that the points

D, E, F are collinear.

Join FD and DE:

then FD and DE shall be in the same

st. line.

Join PB, PC.

Because the Z Â» PDB, PFB are rt. angles, llyp.

:. the points P, D, B, F are concjdic:

.-. the Z PDF=the Z PBF, in the same segment. m. 21.

But since BACP is a quad' inscribed in a circle, having one of its

sides AB produced to F,

.-. the ext. z PBF = the opp. int. z ACP. Ex. 3, p. 188.

.-. the z PDF = the z ACP.

To each add the z PDE :

then the zÂ«PDF, PDE^the z-'ECP, PDE.

But since the z " PDC, PEC are rt. angles,

.'. the points P, D, E, C are concylic ;

.'. the Z â– ECP, PDE together = two rt. angles:

.-. the Z Â» PDF, PDE together = two rt. angles;

.â€¢. FD and DE arc in the same st. line; i. 1-1.

that is, the points D, E, F are collinear. q.e.d.

[This theorem is attributed to Robert Simson; and accordingly

the straight line FDE is sometimes spoken of as the Simson's Line of

the triangle ABC for the point P: some writers also call it the Pedal

of the triangle ABC for the point P.]

75. ABC is a triangle inscribed in a circle ; and from any point P

on the circumference PD, PFare drawn perpendicular to BC and AB:

if FD, or FD produced, outs AC at E, shew that PE is perpendicular

to AC.

76. Pind the locus of a point which moves so that if perpen-

diculars are drawn from it to the sides of a given triangle, their feet

are collinear.

77. ABC and A'B'C are two triangles having a common vertical

angle, and the circles circumscribed about them meet again at P : shew

that the feet of perpendiculars drawn from P to the four lines AB, AC,

BC, B'C are collinear.

THKOUKMS AND EXAMPLES OX IJOOK III.

233

78. A triangle is inscribed in a circle, and any point P on the cir-

cumference is joined to the orthocentre of the triangle : shew that this

joining line is bisected by the pedal of the point P.

OX THE CIRCLE IN CONNECTION WITH RECTAXGLES.

See Propositions 35, 36, 37.

1. If from any external point P tico tangents are draivn to a

given circle tcliose centre is O, and if OP meets the chord of contact

at Ql; then the rectangle OP, OO is equal to the square on the radius.

Let PH, PK be tangents, drawn from

the external point P to the Â© HAK, whose , - -^,

centre is O; and let OP meet HK the

chord of contact at Ql, and the o''^ at A :

then shall the rect. OP, OQ=:the sq. on

OA.

On HP as diameter describe a circle :

this circle must pass through O, since the

Z HOP is a rt. angle. in. 31.

Join OH.

Then since PH is a tangent to the Â© HAK,

. â€¢. the / OHP is a rt. angle.

And since HP is a diameter of the Â© HQP,

. â€¢. OH touches the Â© HOP at H. iii. 16.

.-. the rect. OP, OQ = the sq. on OH, iii. 30.

= the sq. on OA. q. e. j>.

2. ABC is a triangle, and AD, BE, OF the perpendiculars drawn

from the vertices to tlie oj^posite sides, meeting in the orthocentre O :

shew that the rect. AO, OD = tlie rect. BO, OE = the rect. CO, OF.

3. ABC is a triangle, and AD, BE the perpendiculars drawn

from A and B on the opposite sides : shew that the rectangle CA, CE

is equal to the rectangle CB, CD.

4. ABC is a triangle right-angled at C, and from D, any point in

the hypotenuse AB, a straight line DE is drawn perpendicular to AB

and meeting BC at E: shew that the square on DE is equal to

the difference of the rectangles AD, DB and CE, EB.

5. From an external point P two tangents are drawn to a

given circle whose centre is O, and OP meets the chord of contact

at Q: shew that any circle which passes through the points P, Q

will cut the given cii-cle orthogonally. [See Def. p. 222.]

234 EUCLID'S ELEMENTS.

6. A series of circles pass through two i/iven points, and from a

fixed point in the common chord produced tauifcnts are drawn to all the

circles : shew that the points of contact lie on a circle which cuts

all the given circles orthogonally.

7. All circles which pass through a fixed point, and cut a given

circle orthogonally, jxtss also through a second fixed point.

8. Find the locus of the centres of all circles which pass through

a given point and cut a given circle orthogonally.

9. Describe a circle to pass through two given points and cut a

given circle orthogonally.

10. A, B, C, D are four points taken in order on a given straight

line : find a point O between B and C such that the rectangle

OA, OB may be equal to the rectangle OC, OD.

11. AB is a fixed diameter of a circle, and CD a fixed straight

line of indefinite length cutting AB or AB produced at right angles ;

any straight line is drawn through A to cut CD at P and the circle at

Q: sheiv that the rectangle AP, AQ is constant.

12. AB is a fixed diameter of a circle, and CD a fixed chord

at right angles to AB ; any straight line is drawn through A to

cut CD at P and the circle at Q: shew that the rectangle AP, AQ

is equal to the square on AC.

13. A is a fixed point and CD a fixed straight line of indefinite

length; AP is any straight line drawn through A to meet CD at P;

and in AP a point Q is taken such that the rectangle AP, AQ is

constant: find the locus of Q.

14. Two circles intersect orthogonally, and tangents are drawn

from any point on the circumference of one to touch the other:

prove that the first circle passes through the middle point of the

chord of contact of the tangents. [Ex. 1, p. 233.]

15. A semicircle is described on AB as diameter, and any two

chords AC, BD are drawn intersecting at P : shew that

AB2=AC.AP+BD. BP.

16. Two circles intersect at B and C, and the two direct common

tangents AE and DF are drawn : if the common chord is produced to

meet the tangents at G and H, shew that GH- = AE2+ BC^.

17. If from a point P, without a circle, PM is drawn perpendicular

to a diameter AB, and also a secant PCD, shew that

PM2=PC.PDÂ±AM . MB,

according as PM intersects the circle or not.

THEOREMS AND EXAMPJ.ES ON BOOK 111. 235

18. Three circles intersect at D, and their other points of

intersection are A, B, C; AD cuts the circle BDC at E, and EB, EC

cut the circles ADB, ADC respectively at F and G : shew that the

points F, A, G are collinear.

19. A semicircle is described on a given diameter BC, and from

B and C any two chords BE, CF are drawn intersecting within

the semicircle at O; BF and CE are produced to meet at A: shew

that the sum of the squares on AB, AC is equal to twice the square on

the tangent from A together with the square on BC.

20. X and Y are two fixed points in the diameter of a circle

equidistant from the centre C : through X any chord PXQ is drawn,

and its extremities are joined to Y; shew that the sum of the

squares on the sides of the triangle PYQ is constant. [See p. 147,

Ex. 24.]

Pboblems on Tangency.

21. To describe a circle to j[;ass through tico given points and to

touch a given straight line.

Let A and B be the given points,

and CD the given st. line:

it is required to describe a circle to

])ass through A and B and to touch

CD.

Join BA, and produce it to meet

CD at P.

Describe a square equal to the

rect. PA, PB ; ii. 14.

and from PD (or PC) cut off PQ equal to a side of this square.

Through A, B and Q describe a circle. Ex. 4, p. 15G.

Then since the rect, PA, PB = the sq. on PQ,

. â€¢. the ABQ touches CD at Q. iii. 37.

Q. E. F.

Note, (i) Since PQ may be taken on either side of P, it is

clear that there are in general two solutions of the problem.

(ii) When AB is parallel to the given line CD, the above method

is not applicable. In this case a simple construction follows from

iiT. 1- Cor. and iii. 16 â€¢ and it will be found that only one solution

exists

i':iÂ«i

:i.i:m i:n'

2"2. To (h'^crihc a circle lo jms.'; tliroiiyJi tiro tjircit jmints and

to loach (I (jircH circle.

Let A and B be the given

points, and CRP the given

circle :

it is required to describe a

circle to pass through A and

B, and to touch the Â©CRP.

Through A and B de-

scribe any circle to cut the

given circle at P and Q.

Join AB, PQ, and pro- "q

duce them to meet at D.

From D draw DC to touch the given circle, and let C be the point

of contact.

Then the circle described through A, B, C will touch the given

circle.

For, from the Â©ABQP, the rect. DA, DB = the rect. DP, DQ:

and from the PQC, the rect. DP, DQ = the sq. on DC; ki. '.)(',.

.-. the rect, DA, DB = the sq. on DC :

.-. DC touches the Â© ABC at C. in. 37.

But DC totiches the PQC at C ; Constr.

.'. the Â© ABC touches the given circle, and it passes tlirough the

given points A and B. q.k.i'.

Note, (i) Since two tangents may be drawn from D tc the

given circle, it follows that there will be two solutions of the problem.

(ii) The general construction fails when the straight line bisect-

ing AB at right angles passes through the centre of the given circle:

the problem then becomes symmetrical, and the solution is obvious.

23. To describe a circle to pass throiujh a given point and

touch ttco (jiven straight lines.

Let P be the given point, and

AB, AC the given straight lines:

it is required to describe a circle

to j)ass through P and to touch

AB, AC.

Now the centre of every circle

which touches AB and AC must

lie on the bisector of the z BAC,

Ex. 7, p. 183.

Hence draw AE bisecting the

z BAC.

From P draw PK perp. to AE, and produce it to P',

making KP' equal to PK.

THEOREMS AND EXAMPLES ON BOOK III. 237

Then every circle which has its centre in AE, and passes through

P, must also pass through P'. Ex. 1, p. 215.

Hence the problem is now reduced to drawing a circle through

P and P' to touch either AC or AB. Ex. 21, p. 235.

Produce P'P to meet AC at S.

Describe a square equal to the rect. SP, SP'; it. 4 â€¢

and cut off SR equal to a side of the square.

Describe a circle through the points P', P, R:

then since the rect. SP, SP' = the sq. on SR, Con>itr.

.-. the circle touches AC at R ; iii. 37.

and since its centre is in AE, the bisector of the Z BAC,

it may be shewn also to touch AB. q. e. v.

Note, (i) Since SR may be taken on either side of S, it follows

that there will be two solutions of the problem.

(ii) If the given straight lines are parallel, the centre lies on the

parallel straight line mid- way between them, and the construction

proceeds as before.

24. To describe a circle to touch tico given straight lines and a

given circle.

Let AB, AC be the two given H..

st. lines, and D the centre of the ,'^' ^^

given circle : ^\<''' '^^y:

it is required to describe a circle .'-/ \ q ^-^ \

to touch AB, AC and the circle Oy'' ! >0\/'^\N

whose centre is D. f y^\r<\-^^ ' \

Draw EF, GH par' to AB X^^ Vj^ '7^ )

and AC respectively, on the sides ^ \ '^i V / J ^

remote from D, and at distances ..rii-,,]^-.-'^â€”!^!!^!!^

from them equal to the radius of E M ~ F

the given circle.

Describe the Â©MND to touch EF and GH at M and N, and

to pass through D. Ex. 23, p. 23G.

Let O be the centre of this circle.

In like manner it may be proved that

the z DEC = the z FEA=:the z B,

and the Z DFB = the z EFA = the z C.

Corollary, (ii) The triangles DEC, AEF, DBF are equiangular

to one another and to the triangle ABC.

Note. If the angle BAC is obtuse, then the perpendiculars BE, CF

bisect externally the corresponding angles of the pedal triangle.

Ji. E. 15

226 EUCIill/.S ELEMENTS.

21. In a III/ triinifjlr, if the perpendiculars drawn from the vertices

on the opposite s/J, s art' produced to meet the circumscribed circle,

then each side bist'ctn that portion of the line perpendicular to it which

lies between the orthocentre and the circumference.

Let ABC be a triangle in which the perpen-

diculars AD, BE are drawn, intersecting at O the

orthocentre; and let AD be produced to meet

the o*Â® of the circumscribing circle at G :

then shall DO=DG.

Join BG.

Then in the two a" OEA, ODB,

the z OEA = the z ODB, being rt. angles;

and the Z EOA = the vert. opp. Z DOB;

.â€¢. the remaining Z EAO = the remaining Z DBO. i. 32.

But the z CAG=the z CBG, in the same segment;

.-. the Z DBO = the z DBG.

Then in the a" DBO, DBG,

(the Z DBO = the Z DBG, Proved.

Because jthe z BDO = the z BDG,

( and BD is common;

.-. DO = DG. 1.26.

Q. E. D.

22. In an acute-angled triangle the three sides are the external

bisectors of the angles of the pedal triangle : and in an obtuse-angled

triangle the sides containing the obtuse angle are the internal bisectors

of the corresponding angles of the pedal triangle.

23. If O is the orthocentre of the triangle ABC, sheic that the

angles BOC, BAC are supplementary.

24. If O is the orthocentre of the triangle ABC, then any one of

the four points O, A, B, C is the orthocentre of the triangle whose

vertices are the other three.

25. The three circles lohich pass through two vertices of a triangU

and its orthocentre are each equal to the circle circumscribed about the

triangle.

26. D , E are taken on the circumference of a semicircle described

on a given straight line AB : the chords AD, BE and AE, BD

intersect (produced if necessary) at F and G : shew that FG is per-

pendicular to AB.

27. A BCD is a parallelogram; AE and CE are drawn at right

angles to AB, and CB respectively: shew that ED, if produced, will

be perjiendicular to AC.

THEOREMS AND EXAMPLES ON BOOK III. 227

28. ABC is a triangle, O is its orthocentre, and AK a diameter

of the circumscribed circle: shew that BOCK is a parallelogram.

29. The orthocentre of a triangle is joined to the middle point of

the base, and the joining line is produced to meet the circumscribed

circle : prove that it will meet it at the same point as the diameter

which passes through the vertex.

30. The perpendicular from the vertex of a triangle on the base,

and the straight line joining the orthocentre to the middle point of

the base, are produced to meet the circumscribed circle at P and Gt :

shew that PQl is parallel to the base.

31. The distance of each vertex of a triangle from the orthocentre

is double of the perpendicular drawn from the centre of the circum-

scribed circle on the opposite side.

32. Three circles are described each passing through the ortho-

centre of a triangle and two of its vertices: shew that the triangle

formed by joining their centres is equal in all respects to the original

triangle.

33. ABC is a triangle inscribed in a circle, and the bisectors of its

angles which intersect at O are produced to meet the circumference in

PQR : shew that O is the orthocentre of the triangle PQR.

34. Construct a triangle, having given a vertex, the orthocentre,

and the centre of the circumscribed circle.

Loci.

35. Given the base and vertical angle of a triangle, find the locus

of its orthocentre.

Let BC be the given base, and X the

^iven angle ; and let BAC be any triangle

on the base BC, having its vertical z A

equal to the Z X.

Draw the perpÂ« BE, CF, intersecting

at the orthocentre O.

It is required to find the locus of O.

Since the z ' OFA, OEA axe rt. angles,

.-. the points O, F, A, E are concyclic ;

.â€¢.the Z FOE is the supplement of the z A:

.-. the vert. opp. z BOC is the supplement of the Z A.

But the Z A is constant, being always equal to the Z X ;

.â€¢. its supplement is constant ;

that is, the A BOC has a fixed base, and constant vertical angle;

hence the locus of its vertex O is the arc of a segment of which BC is

the chord. [See p. 187.]

15-2

228

KUCLID8 ELEMENTS.

36. Given the base and vertical angle of a triangle, find the luciis

of the intersection of the bisectors of its angles.

Let BAG be any triangle on tlie given

base BC, liaviug its vertical angle equal to

the given z X; and let Al, Bl, CI be the

bisectors of its angles: [see Ex. 2, p. 103.]

it is required to find the locus of the

point I.

Denote the angles of the A ABC by

A, B,C; and let the z BIC be denoted by I.

Then from the a BIC,

(i) l + JB + *C = twort.

and from the a ABC,

A + B + C = two rt. angles ;

(ii) so that ^A + ^B + ^C = one rt. angle,

.". , taking the differences of the equals in (i) and (ii),

I - ^ A == one rt. angle :

l=one rt. angle + ^A.

or,

X;

I. 32.

But A is constant, being always equal to the z

.-. I is constant :

.". , since the base BC is fixed, the locus of I is the arc of a segment

of which BC is the chord.

37. Given the base and vertical angle of a triangle, find the locus

of the centroid, that is, the intersection of the medians.

Let BAC be any triangle on the given

base BC, having its vertical angle equal

to the given angle S; let the medians AX,

BY, CZ intersect at the centroid G [see

Ex. 4, p. 105] :

it is required to find the locus of the point G .

Through G draw GP, GQ par' to AB

and AC respectively.

Then ZG is a third part of ZC;

Ex. 4, p. 105.

and since GP is par' to ZB,

.-. BP is a third part of BC.

Similarly QC is a third part of BC ;

.â€¢. P and Q are fixed points.

Now since PG, GQ are par' respectively to BA, AC,

.-. the z PGQ = the z BAC,

= the z S,

that is, the Z PGQ is constant;

and since the base PQ is fixed,

.-. the locus of G is the arc of a segment of which PQ is the chord.

Ex. 19, p. 09.

Constr.

' I. 29.

THKOIIKMS AND EXAMPLES ON IJOOK III. 229

Ohs. In this problem the points A and G move on the arcs of

similar segments.

38. Given the base and the vertical angle of a triangle ; find the

locus of the intersection of the bisectors of the exterior base angles.

39. Through the extremities of a given straight line AB any two

parallel straight lines AP, BQ are drawn ; find the lociis of the inter-

section of the bisectors of the angles PAB, QBA.

40. Find the locus of the middle points of chords of a circle drawn

through a fixed point.

Distinguish between the cases when the given point is within,

on, or without the circumference.

41. Find the locus of the points of contact of tangents drawn

from a fixed point to a system of concentric circles.

42. Find the locus of the intersection of straight lines which pass

through two fixed points on a circle and intercept on its circumference

an arc of constant length.

43. A and B are two fixed points on the circumference of a circle,

and PQ is any diameter : find the locus of the intersection of PA and

QB.

44. BAG is any triangle described on the fixed base BG and

having a constant vertical angle ; and BA is produced to P, so that

BP is equal to the sum of the sides containing the vertical angle: find

the locus of P.

45. AB is a fixed chord of a circle, and AC is a moveable chord

passing through A: if the parallelogram GB is completed, find the

locus of the intersection of its diagonals.

46. A straight rod PQ slides between two rulers placed at right

angles to one another, and from its extremities PX, QX are drawn

perpendicular to the rulers: find the locus of X.

47. Two circles whose centres are G and D, intersect at A and B :

through A, any straight line PAQ is drawn terminated by the circum-

ferences ; and PG, QD intersect at X: find the locus of X, and shew

that it passes through B. [Ex. 9, p. 216.]

48. Two circles intersect at A and B, and through P, any point

on the circumference of one of them, two straight lines PA, PB

are drawn, and produced if necessary, to cut the other circle at X

and Y: find the locus of the intersection of AY and BX.

49. Two circles intersect at A and B; HAK is a fixed straight

line drawn through A and terminated by the circumferences, and

PAQ is any other straight line similarly drawn: find the locus of the

intersection of HP and QK. [Ex. 3, p. 186.]

::;3() KUtJ.IDS KLKMKNTS.

50. Two segments of circles are on the same chord AB and on

the same side of it ; and P and Q are any points one on each arc :

lintl the locus of the intersection of the bisectors of the angles PAQ,

PBQ.

51. Two circles intersect at A and B ; and through A any straight

line PAQ is drawn terminated by the circumferences : find the locus of

the middle point of PQ.

Miscellaneous Examples on Angles in a Circle.

52. ABC is a triangle, and circles are drawn through B, C, cutting

the sides in P, Q, P', Q', ... : shew that PQ, P'Q' ... are parallel to one

another and to tlie tangent drawn at A to the circle circumscribed

about the triangle.

53. Two circles intersect at B and C, and from any point A, on

the circumference of one of them, AB, AC are drawn, and produced if

necessary, to meet the other at D and E : shew that D E is parallel to

the tangent at A.

64. A secant PAB and a tangent PT are drawn to a circle from

an external point P; and tbe bisector of the angle ATB meets AB at

C : shew that PC is equal to PT.

55. From a point A on the circumference of a circle two chords

AB, AC are drawn, and also the diameter AF: if AB, AC are produced

to meet the tangent at F in D and E, shew that the triangles ABC,

AED are equiangular to one another.

56. O is any point within a triangle ABC, and CD, OE, OF are

drawn j)erpendicular to BC, CA, AB respectively : shew that the

angle BOC is equal to the sum of the angles BAC, EDF.

57. If two tangents are drawn to a circle from an external point,

shew that they contain an angle equal to the difference of the angles

in the segments cut off by the chord of contact.

58. Two circles intersect, and through a point of section a straight

line is drawn bisecting the angle between the diameters through that

point : shew that this straight line cuts off similar segments from the

two circles.

59. Two equal circles intersect at A and B ; and from centre

A, with any radius less than AB a third circle is described cutting the

given circles on the same side of A B at C and D: shew that the points

B, C, D are collinear.

60. ABC and A'B'C are two triangles inscribed in a circle, so that

AB, AC are respectively parallel to A'B', A'C : shew that BC is

parallel to B'C,

THEOREMS AND EXAMPLES ON BOOK III. 231

01. Two circles intersect at A and B, and through A two straight

lines HAK, PAQ are drawn terminated by the circumferences : if

HP and KQ intersect at X, shew that the points H, B, K, X are

concyclic.

62. Describe a circle touching a given straight line at a given

point, so that tangents drawn to it from two fixed points in the given

line may be parallel. [See Ex. 10, p. 183.]

G3. C is the centre of a circle, and CA, CB two fixed radii: if

from any point P on the arc AB perpendiculars PX, PY are drawn to

CA and CB, shew that the distance XY is constant.

64. AB is a chord of a circle, and P any point in its circum-

ference ; PM is drawn perpendicular to AB, and AN is drawn perpen-

dicular to the tangent at P : shew that MN is parallel to PB.

65. P is any point on the circumference of a circle of which AB is

a fixed diameter, and PN is drawn perpendicular to AB ; on AN and

BN as diameters circles are described, which are cut by AP, BP

at X and Y : shew that XY is a common tangent to these circles;

66. Upon the same chord and on the same side of it three seg-

ments of circles are described containing respectively a given angle,

its supplement and a right angle: shew that the intercept made by the

two former segments upon any straight line drawn through an ex-

tremity of the given chord is bisected by the latter segment.

67. Two straight lines of indefinite length touch a given circle,

and any chord is drawn so as to be bisected by the chord of contact :

if the former chord is produced, shew that the intercepts between the

circumference and the tangents are equal.

68. Two circles intersect one another: through one of the points

of contact draw a straight line of given length terminated by the cir-

cumferences.

69. On the three sides of any triangle equilateral triangles are

described remote from the given triangle : shew that the circles de-

scribed about them intersect at a point.

70. On BC, CA, AB the sides of a triangle ABC, any points

P, Q, R are taken; shew that the circles described about the triangles

AQR, BRP, CPQ meet in a point.

71. Find a point within a triangle at which the sides subtend

equal angles.

72. Describe an equilateral triangle so that its sides may pass

through three given points.

73. Describe a triangle equal in all respects to a given triangle,

and having its sides passing through three given points.

â€¢2:i-2 KUCLID'b KLKMENTS.

Simbon'b Line.

74. If from any point on the circumference of the circle circum-

scribed about a triangle, pei-pendiculars are drawn to the three siden, the

feet of these perpendiculars are coUinear.

Let P be any point on the o"" of the

circle circumscribed about the a ABC ;

and let PD, PE, PF be the perp" drawn

from P to the three sides.

It is required to prove that the points

D, E, F are collinear.

Join FD and DE:

then FD and DE shall be in the same

st. line.

Join PB, PC.

Because the Z Â» PDB, PFB are rt. angles, llyp.

:. the points P, D, B, F are concjdic:

.-. the Z PDF=the Z PBF, in the same segment. m. 21.

But since BACP is a quad' inscribed in a circle, having one of its

sides AB produced to F,

.-. the ext. z PBF = the opp. int. z ACP. Ex. 3, p. 188.

.-. the z PDF = the z ACP.

To each add the z PDE :

then the zÂ«PDF, PDE^the z-'ECP, PDE.

But since the z " PDC, PEC are rt. angles,

.'. the points P, D, E, C are concylic ;

.'. the Z â– ECP, PDE together = two rt. angles:

.-. the Z Â» PDF, PDE together = two rt. angles;

.â€¢. FD and DE arc in the same st. line; i. 1-1.

that is, the points D, E, F are collinear. q.e.d.

[This theorem is attributed to Robert Simson; and accordingly

the straight line FDE is sometimes spoken of as the Simson's Line of

the triangle ABC for the point P: some writers also call it the Pedal

of the triangle ABC for the point P.]

75. ABC is a triangle inscribed in a circle ; and from any point P

on the circumference PD, PFare drawn perpendicular to BC and AB:

if FD, or FD produced, outs AC at E, shew that PE is perpendicular

to AC.

76. Pind the locus of a point which moves so that if perpen-

diculars are drawn from it to the sides of a given triangle, their feet

are collinear.

77. ABC and A'B'C are two triangles having a common vertical

angle, and the circles circumscribed about them meet again at P : shew

that the feet of perpendiculars drawn from P to the four lines AB, AC,

BC, B'C are collinear.

THKOUKMS AND EXAMPLES OX IJOOK III.

233

78. A triangle is inscribed in a circle, and any point P on the cir-

cumference is joined to the orthocentre of the triangle : shew that this

joining line is bisected by the pedal of the point P.

OX THE CIRCLE IN CONNECTION WITH RECTAXGLES.

See Propositions 35, 36, 37.

1. If from any external point P tico tangents are draivn to a

given circle tcliose centre is O, and if OP meets the chord of contact

at Ql; then the rectangle OP, OO is equal to the square on the radius.

Let PH, PK be tangents, drawn from

the external point P to the Â© HAK, whose , - -^,

centre is O; and let OP meet HK the

chord of contact at Ql, and the o''^ at A :

then shall the rect. OP, OQ=:the sq. on

OA.

On HP as diameter describe a circle :

this circle must pass through O, since the

Z HOP is a rt. angle. in. 31.

Join OH.

Then since PH is a tangent to the Â© HAK,

. â€¢. the / OHP is a rt. angle.

And since HP is a diameter of the Â© HQP,

. â€¢. OH touches the Â© HOP at H. iii. 16.

.-. the rect. OP, OQ = the sq. on OH, iii. 30.

= the sq. on OA. q. e. j>.

2. ABC is a triangle, and AD, BE, OF the perpendiculars drawn

from the vertices to tlie oj^posite sides, meeting in the orthocentre O :

shew that the rect. AO, OD = tlie rect. BO, OE = the rect. CO, OF.

3. ABC is a triangle, and AD, BE the perpendiculars drawn

from A and B on the opposite sides : shew that the rectangle CA, CE

is equal to the rectangle CB, CD.

4. ABC is a triangle right-angled at C, and from D, any point in

the hypotenuse AB, a straight line DE is drawn perpendicular to AB

and meeting BC at E: shew that the square on DE is equal to

the difference of the rectangles AD, DB and CE, EB.

5. From an external point P two tangents are drawn to a

given circle whose centre is O, and OP meets the chord of contact

at Q: shew that any circle which passes through the points P, Q

will cut the given cii-cle orthogonally. [See Def. p. 222.]

234 EUCLID'S ELEMENTS.

6. A series of circles pass through two i/iven points, and from a

fixed point in the common chord produced tauifcnts are drawn to all the

circles : shew that the points of contact lie on a circle which cuts

all the given circles orthogonally.

7. All circles which pass through a fixed point, and cut a given

circle orthogonally, jxtss also through a second fixed point.

8. Find the locus of the centres of all circles which pass through

a given point and cut a given circle orthogonally.

9. Describe a circle to pass through two given points and cut a

given circle orthogonally.

10. A, B, C, D are four points taken in order on a given straight

line : find a point O between B and C such that the rectangle

OA, OB may be equal to the rectangle OC, OD.

11. AB is a fixed diameter of a circle, and CD a fixed straight

line of indefinite length cutting AB or AB produced at right angles ;

any straight line is drawn through A to cut CD at P and the circle at

Q: sheiv that the rectangle AP, AQ is constant.

12. AB is a fixed diameter of a circle, and CD a fixed chord

at right angles to AB ; any straight line is drawn through A to

cut CD at P and the circle at Q: shew that the rectangle AP, AQ

is equal to the square on AC.

13. A is a fixed point and CD a fixed straight line of indefinite

length; AP is any straight line drawn through A to meet CD at P;

and in AP a point Q is taken such that the rectangle AP, AQ is

constant: find the locus of Q.

14. Two circles intersect orthogonally, and tangents are drawn

from any point on the circumference of one to touch the other:

prove that the first circle passes through the middle point of the

chord of contact of the tangents. [Ex. 1, p. 233.]

15. A semicircle is described on AB as diameter, and any two

chords AC, BD are drawn intersecting at P : shew that

AB2=AC.AP+BD. BP.

16. Two circles intersect at B and C, and the two direct common

tangents AE and DF are drawn : if the common chord is produced to

meet the tangents at G and H, shew that GH- = AE2+ BC^.

17. If from a point P, without a circle, PM is drawn perpendicular

to a diameter AB, and also a secant PCD, shew that

PM2=PC.PDÂ±AM . MB,

according as PM intersects the circle or not.

THEOREMS AND EXAMPJ.ES ON BOOK 111. 235

18. Three circles intersect at D, and their other points of

intersection are A, B, C; AD cuts the circle BDC at E, and EB, EC

cut the circles ADB, ADC respectively at F and G : shew that the

points F, A, G are collinear.

19. A semicircle is described on a given diameter BC, and from

B and C any two chords BE, CF are drawn intersecting within

the semicircle at O; BF and CE are produced to meet at A: shew

that the sum of the squares on AB, AC is equal to twice the square on

the tangent from A together with the square on BC.

20. X and Y are two fixed points in the diameter of a circle

equidistant from the centre C : through X any chord PXQ is drawn,

and its extremities are joined to Y; shew that the sum of the

squares on the sides of the triangle PYQ is constant. [See p. 147,

Ex. 24.]

Pboblems on Tangency.

21. To describe a circle to j[;ass through tico given points and to

touch a given straight line.

Let A and B be the given points,

and CD the given st. line:

it is required to describe a circle to

])ass through A and B and to touch

CD.

Join BA, and produce it to meet

CD at P.

Describe a square equal to the

rect. PA, PB ; ii. 14.

and from PD (or PC) cut off PQ equal to a side of this square.

Through A, B and Q describe a circle. Ex. 4, p. 15G.

Then since the rect, PA, PB = the sq. on PQ,

. â€¢. the ABQ touches CD at Q. iii. 37.

Q. E. F.

Note, (i) Since PQ may be taken on either side of P, it is

clear that there are in general two solutions of the problem.

(ii) When AB is parallel to the given line CD, the above method

is not applicable. In this case a simple construction follows from

iiT. 1- Cor. and iii. 16 â€¢ and it will be found that only one solution

exists

i':iÂ«i

:i.i:m i:n'

2"2. To (h'^crihc a circle lo jms.'; tliroiiyJi tiro tjircit jmints and

to loach (I (jircH circle.

Let A and B be the given

points, and CRP the given

circle :

it is required to describe a

circle to pass through A and

B, and to touch the Â©CRP.

Through A and B de-

scribe any circle to cut the

given circle at P and Q.

Join AB, PQ, and pro- "q

duce them to meet at D.

From D draw DC to touch the given circle, and let C be the point

of contact.

Then the circle described through A, B, C will touch the given

circle.

For, from the Â©ABQP, the rect. DA, DB = the rect. DP, DQ:

and from the PQC, the rect. DP, DQ = the sq. on DC; ki. '.)(',.

.-. the rect, DA, DB = the sq. on DC :

.-. DC touches the Â© ABC at C. in. 37.

But DC totiches the PQC at C ; Constr.

.'. the Â© ABC touches the given circle, and it passes tlirough the

given points A and B. q.k.i'.

Note, (i) Since two tangents may be drawn from D tc the

given circle, it follows that there will be two solutions of the problem.

(ii) The general construction fails when the straight line bisect-

ing AB at right angles passes through the centre of the given circle:

the problem then becomes symmetrical, and the solution is obvious.

23. To describe a circle to pass throiujh a given point and

touch ttco (jiven straight lines.

Let P be the given point, and

AB, AC the given straight lines:

it is required to describe a circle

to j)ass through P and to touch

AB, AC.

Now the centre of every circle

which touches AB and AC must

lie on the bisector of the z BAC,

Ex. 7, p. 183.

Hence draw AE bisecting the

z BAC.

From P draw PK perp. to AE, and produce it to P',

making KP' equal to PK.

THEOREMS AND EXAMPLES ON BOOK III. 237

Then every circle which has its centre in AE, and passes through

P, must also pass through P'. Ex. 1, p. 215.

Hence the problem is now reduced to drawing a circle through

P and P' to touch either AC or AB. Ex. 21, p. 235.

Produce P'P to meet AC at S.

Describe a square equal to the rect. SP, SP'; it. 4 â€¢

and cut off SR equal to a side of the square.

Describe a circle through the points P', P, R:

then since the rect. SP, SP' = the sq. on SR, Con>itr.

.-. the circle touches AC at R ; iii. 37.

and since its centre is in AE, the bisector of the Z BAC,

it may be shewn also to touch AB. q. e. v.

Note, (i) Since SR may be taken on either side of S, it follows

that there will be two solutions of the problem.

(ii) If the given straight lines are parallel, the centre lies on the

parallel straight line mid- way between them, and the construction

proceeds as before.

24. To describe a circle to touch tico given straight lines and a

given circle.

Let AB, AC be the two given H..

st. lines, and D the centre of the ,'^' ^^

given circle : ^\<''' '^^y:

it is required to describe a circle .'-/ \ q ^-^ \

to touch AB, AC and the circle Oy'' ! >0\/'^\N

whose centre is D. f y^\r<\-^^ ' \

Draw EF, GH par' to AB X^^ Vj^ '7^ )

and AC respectively, on the sides ^ \ '^i V / J ^

remote from D, and at distances ..rii-,,]^-.-'^â€”!^!!^!!^

from them equal to the radius of E M ~ F

the given circle.

Describe the Â©MND to touch EF and GH at M and N, and

to pass through D. Ex. 23, p. 23G.

Let O be the centre of this circle.

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