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,eircles taken in pairs is called the radical centre.



THEOREMS AND EXAMPLES ON BOOK VI.



373



rt. To draw the radical axis of two given circles.




Let A and B be the centres of the given circles :
it is required to draw their radical axis.

If the given circles intersect, then the st. line drawn through their
points of intersection will be the radical axis. [Ex. 1, Cor. p. 372.]
But if the given circles do not intersect,

describe any circle so as to cut them in E, F and G , H :
Join EF and HG, and produce them to meet in P.

Join AB; and from P draw PS perp. to AB.
Then PS shall be the radical axis of the ©" (A), (B).



Definition. If each pair of circles in a given system have
the same radical axis, the circles are said to be co-axal.



EXAMPLES.



1. Shew that the radical axis of two circles bisects any one of their
common tangents.

2. If tangents are drawn to two circles from any point on their
radical axis; sheic that a circle described with this point as centre and
any one of the tangents as radius, exits both the given circles ortho-
gonally.

3. O is the radical centre of three circles, and from O a tangent
OT is draion to any one of them: shew that a circle ivhose centre is O
and radius OT cuts all the given circles orthogonally.

4. If three circles touch one another, taken two and two, shew that
their common tangents at the points of contact are concurrent.



374 EUCLIDS KLEMENTS.

5. If circles are described on the three sides of a triangle as
dia7)ieter, their radical centre is the orthocentre of the triangle.

6. All circles which pass through a fixed point and cut a given
circle orthogonally, pass through a second fixed point.

7. Find the locus of tlie centres of all circles which pass through a
given point and cut a given circle orthogonally.

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

9. Find the locus of the centres of all circles which ciit two given
circles orthogonally.

10. Describe a circle to pass through a given point and cut two
given circles orthogonally.

11. The difference of the squares on the tangents drawn from any
point to two circles is equal to twice the rectangle contained by the
straight line joining their centres and the perpendicular from the givin
point on their radical axis.

12. In a system of co-axal circles which do not intersect, any point
is taken on the radical axis-; shew that a circle described from this
point as centre with radius equal to the tangent draionfromit to any
one of the circles, will meet the line of centres in tioo fixed points.

[These fixed points are called the Limiting Points of the system.'^

13. In a system of co-axal circles the tioo limiting points and the
points in which any one circle of the system cuts the line of centres
form a harmonic range.

14. In a system of co-aocal circles a limiting point has the same
polar with regard to all the circles of the system.

15. If two circles are orthogonal any diameter of one is cut
Jiarmonically by the other.



Obs. In the two following theorems we are to suppose that
the segments of straight lines are expressed numerically in
terms of some common unit ; and the ratio of one such segment
to another will be denoted by the fraction of v/hich the lirst is
the numerator and the second the denominator.



THEOREMS AND EXAMPLES ON BOOK VT. 375



V. ON TRANSVERSALS.

Definition. A straight line drawn to cut a given system of
lines is called a transversal.

1. If three concurrent straigJu lines are draicn from the ancpdar
points of a triangle to meet the apposite sides, then the product of three
alternate segments taken in order is equal to the product of the other
tliree segments.

F





B D C B CD



Let AD, BE, CF be drawn from the vertices of the a ABC to
intersect at O, and cut the opposite sides at D, E, F:
then shall BD . CE . AF^ DC . EA . FB.

By similar triangles it may be shewn that

BD : DC = the alt. of aAOB : the alt. of a AOC;

BD_ aAOB

■ DC ~ aAOC '

. ., , CE aBOC

simdarly, EA^aBOA'

AF aCOA
'•^"^^ FB==ZCOB-

Multiplying these ratios, we have

BD CE AF_,.

DC • EA • FB~ *

or, CD.CE.AF^DC.EA.FB. q. k. n.

The converse of this theorem, which may be proved indirectly, is
very important : it may be enunciated thus :

If three straight lines drawn from the vertices of a triangle cut the
opposite sides so that the product of three alternate segments taken in
order is equal to the product of the other three, then the three straight
lines are concurrent.

That is, if BD . CE . AF= DC . EA . FB,
then AD, BE, CF are concurrent.



;i7H



KUCLIDS KLKMKNTS.



2. If a transversal is drawn to cut tlie sides, or the sides produced,
of a triangle, the product of three alternate segments taken in order is
equal to the product of the other three segments.




T^et ABC be a triangle, and let a transversal meet the sides BC.
CA, AB, or these sides produced, at D, E, F:

then shall BD . CE . AF= DC . EA . FB.
Draw AH par' to BC, meeting the transversal at H.



'Hien from the similar a^ DBF, HAF,
BDHA
FB " AF •
DCE, HAE,
CEEA
DC ~ HA '



and from the similar a '



, by multiplication, ^^
F B



that is.



or,



CEEA
DC ~ AF '
BD^CE^AF
DCTEA . FB~ '

BD.CE. AF-DC



EA . FB.



Q.E. D.

Note. In this theorem the transversal must either meet two
sides and the third side produced, as in Fig. 1; or all three sides pro-
duced, as in Fig. 2.

The converse of this Theorem may be proved indirectly :

If three points are taken in txco sides of a triangle and the third
side produced, or in all three sides produced, so that the product of
three alternate segments taken in order is equal to the product of the
other three segments, the three points are collinear.

The propositions given on pages 103 — 106 relating to the concur-
rence of straight lines in a triangle, may be proved by the method of
Iransversals, and in addition to these the following important theorems
may be established.



MISCELLANEOUS EXAMPLES ON BOOK VI. 377

DEFINITIONS.

(i) If two triangles are such that three straight lines joining
corresponding vertices are concurrent, they are said to be co-
polar.

(ii) If two triangles are such that the points of intersection
of corresponding sides are collinear, they are said to be co-axial.

Theorems to be proved by Transversals.

1. The straight lines which join the vertices of a triangle to the
points of contact of the inscribed circle {or any of the three inscrihcd
circles) are concurrent.

2. The middle points of the diagonals of a complete quadrilateral
are collinear.

3. Co-polar triangles are also co-axial; and conversely co-axial
triangles are also co-polar.

4. The six centres of siviilitude of three circles lie three by three
on four straight lines.



MISCELLANEOUS EXAMPLES ON BOOK VI.



1. Through D, any point in the base of a triangle ABC,
straight lines DE, DF are drawn parallel to the sides AB, AC, and
meeting the sides at E, F: shew that the triangle AEF is a mean
proportional between the triangles FBD, EDC.

2. If two triangles have one angle of the one equal to one
angle of the other, and a second angle of the one supplementary to a
second angle of the other, then the sides about the third angles are
proportional.

3. AE bisects the vertical angle of the triangle ABC and meets
the base in E ; shew that if circles are described about the triangles
ABE, ACE, the diameters of these circles are to each other in the
same ratio as the segments of the base.

4. Through a fixed point O draw a straight line so that the
parts intercepted between O and the perpendiculars drawn to the
straight line from two other fixed points may have a given ratio.



• >/^ KtlCI.IDS KI.KMKNTS.

5. The angle A of a triangle ABC is bisected by AD meeting
BC in D, and AX is the median bisecting BC: shew that XD has
the same ratio to XB as the difference of the sides has to their sum.

6. AD and AE bisect the vertical angle of a triangle internally
and externally, meeting the base in D and E ; shew that if O is the
middle point of BC, then OB is a mean proportional between OD
and OE.

7. P and Q are fixed points; AB and CD are fixed parallel
straight lines; any straight line is drawn from P to meet AB at M,
and a straight line is drawn from Q parallel to PM meeting CD at
N : shew that the ratio of PM to QN is constant, and thence shew
that the straight line through M and N passes through a fixed point.

8. C is the middle point of an arc of a circle whose chord is
AB ; D is any point in the conjugate arc : shew that

AD + DB : DC :: AB : AC.

9. In the triangle ABC the side AC is double of BC. If CD,
CE bisect the angle ACB internally and externally meeting AB in D
and E, shew that the areas of the triangles CBD, ACD, ABC, CDE
are as 1, 2, 3, 4.

10. AB, AC are two chords of a circle; a line parallel to the
tangent at A cuts AB, AC in D and E respectively: shew that the
rectangle AB, AD is equal to the rectangle AC, AE.

11. If from any point on the hypotenuse of a right-angled
triangle perpendiculars are drawn to the two sides, the rectangle
contained by the segments of the hypotenuse will be equal to the
sum of the rectangles contained by the segments of the sides.

12. D is a point in the side AC of the triangle ABC, and E is a
point in AB. If BD, CE divide each other into parts in the ratio
•i : 1, then D, E divide CA, BA in the ratio 3:1.

13. If the perpendiculars from two fixed points on a straight
line passing between them be in a given ratio, the straight line must
pass through a third fixed point.

14. PA, PB are two tangents to a circle; PCD any chord through
P : shew that the rectangle contained by one pair of opposite sides of
the quadrilateral ACBD is equal to the rectangle contained by the
other pair.

15. A, B, C are any three points on a circle, and the tangent at
A meets BC produced in D : shew that the diameters of the circles
circumscribed about A BD, ACD are as AD to CD.



MISCELLANEOUS EXAMPLES ON BOOK VI. 379

16. AB, CD arft two diameters of the circle ADBC at right angles
to each other, and EF is any chord; CE, CF are drawn meeting AB
produced in G and H : prove that the rect. CE, HG = the rect. EF, CH.

17. From the vertex A of any triangle ABC draw a line meeting
BC produced in D so that AD may be a mean proportional between
the segments of the base.

18. Two circles touch internally at O; AB a chord of the larger
circle touches the smaller in C which is cut by the lines OA, OB in
the points P, Q: shew that OP : OQ : : AC : CB.

19. AB is any chord of a circle; AC, BC are drawn to any
point C in the circumference and meet the diameter perpendicular to
AB at D, E : if O be the centre, shew that the rect. OD, OE is equal
to the square on the radius.

20. YD is a tangent to a circle drawn from a point Y in the
diameter AB produced; from D a perpendicular DX is drawn to the
diameter: shew that the points X, Y divide AB internally and ex-
ternally in the same ratio.

21. Determine a point in the circumference of a circle, from
which lines drawn to two other given points shall have a given ratio.

22. O is the centre and OA a radius of a given circle, and V
is the middle point of OA ; P and Q are two points on the circum-
ference on opposite sides of A and equidistant from it; QV is pro-
duced to meet the circle in L : shew that, whatever be the length of
the arc PQ, the chord LP will always meet OA produced in a fixed
point.

23. EA, EA' are diameters of two circles touching each other
externally at E ; a chord AB of the former circle, when produced,
touches the latter at C, while a chord A'B of the latter touches the
former at C : prove that the rectangle, contained by AB and A'B', is
four times as great as that contained by BC and B'C.

24. If a circle be described touching externally two given circles,
the straight line passing through the points of contact will intersect
the line of centres of the given circles at a fixed point.

25. Two circles touch externally in C ; if any point D be taken
without them so that the radii AC, BC subtend equal angles at D,
and DE, DF be tangents to the circles, shew that DC is a mean
proportional between DE and DF.



380 Euclid's elements.

26. If through the middle point of the base of a triangle any
line be drawn intersecting one side of the triangle, the other produced,
and the line drawn parallel to the base from the vertex, it will be
divided harmonically.

27. If from either base angle of a triangle a line be drawn
intersecting the median from the vertex, the opposite side, and the
line drawn parallel to the base from the vertex, it will be divided
harmonically.

28. Any straight line drawn to cut the arms of an angle and its
internal and external bisectors is cut harmonically.

29. P, Q are harmonic conjugates of A and B, and C is an
external point : if the angle PCQ is a right angle, shew that CP, CQ
are the internal and external bisectors of the angle ACB.

30. From C, one of the base angles of a triangle, draw a straight
line meeting AB in G, and a straight line through A parallel to the
base in E, so that CE may be to EG in a given ratio.

31. P is a given point outside the angle formed by two given lines
AB, AC: shew how to draw a straight line from P such that the
parts of it intercepted between P and the lines AB, AC may have a
given ratio.

32. Through a given point within a given circle, draw a straight
line such that the parts of it intercepted between that point and the
circumference may have a given ratio. How many solutions does
the problem admit of? .

33. If a common tangent be drawn to any number of circles
which touch each other internally, and from any point of this
tangent as a centre a circle be described, cutting the other circles ;
and if from this centre lines be drawn through the intersections of
the circles, the segments of the lines within each circle shall be equal.

34. APB is a quadrant of a circle, SPT a line touching it at P;
C is the centre, and PM is perpendicular to CA: prove that

the A SCT : the a ACB :: the a ACB : the a CMP.

35. ABC is a triangle inscribed in a circle, AD, AE are lines
drawn to the base BC parallel to the tangents at B, C respectively ;
shew that AD = AE, and BD : CE :: AB^ : AC^.

36. AB is the diameter of a circle, E the middle point of the
radius OB; on AE, EB as diameters circles are described; PQL is a
common tangent meeting the circles at P and Q, and AB produced
at L: shew that BL is equal to the radius of the smaller circle.



MISCELLANEOUS EXAMPLES ON BOOK VI. 381

37. The vertical angle C of a triangle is bisected by a straight
line which meets the base at D, and is produced to a point E, such
that the rectangle contained by CD and CE is equal to the rectangle
contained by AC and CB: shew that if the base and vertical angle
be given, the position of E is invariable.

38. ABC is an isosceles triangle having the base angles at B
and C each double of the vertical angle: if BE and CD bisect the
base angles and meet the opposite sides in E and D, shew that DE
divides the triangle into figures whose ratio is equal to that of AB
to BC.

39. If AB, the diameter of a semicircle, be bisected in C and on
AC and CB circles be described, and in the space between the three
circumferences a circle be inscribed, shew that its diameter will be
to that of the equal circles in the ratio of two to three.

40. O is the centre of a circle inscribed in a quadrilateral A BCD ;
a line EOF is drawn and making equal angles with AD and BC, and
meeting them in E and F respectively : shew that the triangles A EC,
BOF are similar, and that

AE : ED = CF : FB.

41. From the last exercise deduce the following: The inscribed
circle of a triangle ABC touches AB in F; XOY is drawn through
the centre making equal angles with AB and AC, and meeting them
in X and Y respectively: shew that BX : XF=r AY : YC.

42. Inscribe a square in a given semicircle.

43. Inscribe a square in a given segment of a circle.

44. Describe an equilateral triangle equal to a given isosceles
triangle.

45. Describe a square having given the difference between a
diagonal and a side.

46. Given the vertical angle, the ratio of the sides containing it,
and the diameter of the circumscribing circle, construct the triangle.

47. Given the vertical angle, the line bisecting the base, and the
angle the bisector makes with the base, construct the triangle.

48. In a given circle inscribe a triangle so that two sides may
pass through two given points and the third side be parallel to a
given straight line.

49. In a given circle inscribe a triangle so that the sides may
pass through three given points.



382 EUCLID'S ELEMENTS.

50. A, B, X, Y are four points in a straight line, and O is such
a point in it that the rectangle OA, OB is equal to the rectangle OX,
OY: if a circle be described with centre O and radius equal to a
mean proportional between OA and OB, shew that at every point on
this circle AB and XY will subtend equal angles.

51. O is a fixed point, and OP is any line drawn to meet a fixed
straight line in P; if on OP a point Q is taken so that OQ to OP is
a constant ratio, find the locus of Q.

52. O is a fixed point, and OP is any line drawn to meet the
circumference of a fixed circle in P; if on OP a point Q is taken so
that OQ to OP is a constant ratio, find the locus of Q.

53. If from a given point two straight lines are drawn including
a given angle, and having a fixed ratio, find the locus of the extremity
of one of them when the extremity of the other lies on a fixed straight
line. •

54. On a straight line PAB, two points A and B are marked and
the line PAB is made to revolve round the fixed extremity P. is a
fixed point in the plane in which PAB revolves; prove that if CA
and CB be joined and the parallelogram CADB be completed, the
locus of D will be a circle.

55. Find the locus of a point whose distances from two fixed
points are in a given ratio.

56. Find the locus of a point from which two given circles sub-
tend the same angle.

57. Find the locus of a point such that its distances from two
intersecting straight lines are in a given ratio.

58. In the figure on page 364, shew that QT, P'T' meet on the
radical axis of the two circles.

59. Through two given points draw a circle cutting another
circle so that their common chord may be equal to a given straight
line.

60. ABC is any triangle, and on its sides equilateral triangles
are described externally : if X, Y, Z are the centres of their inscribed
circles, shew that the triangle XYZ is equilateral.



SOLID GEOMETEY.



EUCLID. BOOK XL



Definitions.

From tlie Definitions of Book I. it will be remembered
that

(i) A line is that which has length, without breadth
or thickness.

(ii) A surface is that which has length and breadth,
without thickness.

To these definitions we have now to add :

(iii) Space is that which has length, hreadth, and
thickness.

Thus a line is said to be of one dimension ;

a surface is said to be of two dimensions ;
and space is said to be of three dimensions.

The Propositions of Euclid's Eleventh Book here given
establish the first principles of the geometry of space, or
solid geometry. They deal with the properties of straight
lines which are not all in the same plane, the relations
which straight lines bear to planes which do not contain
those lines, and the relations which two or more planes
bear to one another. Unless the contrary is stated the
straight lines are supposed to be of indefinite length, and
the planes of infinite extent.

Solid geometry then proceeds to discuss the properties
of solid figures, of surfaces which are not planes, and of
lines which can not be drawn on a plane surface.



384



EUCLID b ELEMENTS.

Lines and Planes.



1. A straight line is perpendicular to a plane when
it is perpendicular to every straight* line which meets it
in that plane.



^






Note. It will be proved in Proposition 4 that if a straight line
is perpendicular to two straight lines which meet it in a plane, it is
also perpendicular to every straight line which meets it in that plane.

A straight line drawn perpendicular to a plane is said to be a
normal to that plane.

2. The foot of the perpendicular let fall from a given
point on a plane is called the projection of that point on
the plane.

3. The projection of aline on a plane is the locus of
the feet of perpendiculars drawn from all points in the
given line to the plane.




Thus in the above ligure the line ah is the projection of the line
AB on the plane PQ.

It will be proved hereafter (sec page 420) that the projection of a
straight line on a plane is also a straight line.



DEFINITIONS.



385



4. The inclination of a straight line to a plane is the
acute angle contained by that line and another drawn from
the point at which the first line meets the plane to the
point at which a perpendicular to the plane let fall from
any point of the first line meets the plane.




Thus in the above figure, if from any point X in the given
straight line AB, which intersects the plane PQ at A, a perpen-
dicular Xx is let fall on the plane, and the straight line Axb is drawn
from A through x, then the inclination of the straight line AB to the
plane PQ is measured by the acute angle BA6. In other words : —

The inclination of a straight line to a plane is the acute angle
contained by the given straight line and its projection on the plane.



Axiom. If two surfaces intersect
meet in a line or lines.



another, they



5. The common section of two intersecting surfaces
is the line (or lines) in which they meet.




Note. It is proved in Proposition 3 that the common section of
two planes is a straight line.

Thus AB, the common section of the two planes PQ, XY is proved
to be a straight line.



386



EUCLID'a ELEMENTS.



6. One plane is perpendicular to another plane when
any straight line drawn in one of the planes perpendicular
to the common section is also perpendicular to the other
plane.



•^ (BimiiiMisif iiiiiillli^



A..^




iilliilliliiiliiQilillliiliiiliilliliiiiiillli^^^




Thus in the adjoining figure, the plane EB is perpendicular to the
plane CD, if any straight line PQ, drawn in the plane EB at right
angles to the common section AB, is also at right angles to the
plane CD.

7. The inclination of a plane to a plane is the acute
angle contained by two straight lines drawn from any point
in the common section at right angles to it, one in one
plane and one in the other.



Thus in the adjoining figure,
the straight line AB is the com-
mon section of the two inter-
secting planes BC, AD; and
from Q, any point in AB, two
straight lines QP, QR are drawn
perpendicular to AB, one in each
plane: then the inclination of
the two planes is measured by
the acute angle PQR.



Note. This definition assumes that the angle PQR is of constant
magnitude whatever point Ql is taken in AB : the truth of which
assumption is proved in Proposition 10.

The angle formed by the intersection of two planes is called a
dihedral angle.

It may be proved tljut two planes are perpendicular to one another
when the dihedral angle formed by them is a right angle.




DEFINITIONS.



387



8. Parallel planes are such as do not meet when pro-
duced.

9. A straight line is parallel to a plane if it does not
meet the plane when produced.

10. The angle between two straight lines which do not
meet is the angle contained by two intersecting straight
lines respectively parallel to the two non-intersecting lines.



Thus if AB and CD are two
straight hues which do not meet,
and ah, he are two intersecting lines
parallel respectively to AB and CD ;
then the angle between AB and CD
is measured by the angle abc.




11. A solid angle is that which is made by three or
more plane angles which have a common vertex, but are
not in the same plane.



A solid angle made by three
plane angles is said to be trihedral ;
if made by more than three, it is
said to be polyhedral.

A sohd angle is sometimes called
a comer.




12. A solid figure is any portion of space bounded by
bne or more surfaces, plane or curved.

These surfaces are called the faces of the solid, and the inter-
sections of adjacent faces are called edges.

25-2



388



EUCLID S ELEMENTS.



POLVHKDRA.

13. A polyhedron is a solid figure bounded by plane
faces.

Obs. A plane rectilineal figure must at least have three sides;
or four, if two of the sides are parallel. A polyhedron must at least


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