Page images
PDF
EPUB
[ocr errors]

P

[ocr errors]

C'B'

333o. Let A, A', B, B', C, C' be six points in involution, and let O be the centre,

Draw any line OPQ through 0, and take P and Q so that

OP.OQ=OA, OA', and join PA, PB, PC, and PC', and also QA', QB, QC, and QC. Then, ::

A'

A B
OA.OA=OP.OQ,

A, P, Q, A' are concyclic... LOPA=LOA'Q. Similarly, B, P, Q, B’ are concyclic, and LOPB=LOB’Q, etc.

LAPB=LA'QB. Similarly, LBPC=_B'QC, LCPC'=LC'QC, etc. Hence the pencils P(ABCC') and Q(A'B'C'C) are equianharmonic, or ABCC'} = {A'B'C'C}. Hence also {ABB‘C} = A'B'BC'}, {AA’BC} = {A’AB'C'}. And any one of these relations expresses the condition that the six points symbolized may be in involution.

334o. As involution is only a species of homography, the relations constantly existing between homographic ranges and their corresponding pencils, hold also for ranges and pencils in involution. Hence

1. Every range in involution determines a pencil in involution at every vertex, and conversely.

2. If a range in involution be projected rectilinearly through any point on a circle it determines a system in involution on the circle, and conversely.

Ex. The three pairs of opposite connectors of any four points cut any line in a six-point involution.

A, B, C, D are the four points,
and P, P' the line cut by the six
connectors CD, DA, AC, CB, BD,
and AB. Then
D{PQRR'S = D{CARB}

=BCARD
=BQ'P'RR'} = P'Q'R'R},

(302)

P

D

R

А

P

B

[ocr errors]

T

{PQRR'} = {P'Q'R'R}, and the six points are in involution.

Cor. 1. The centre o of the involution is the radical centre of any three circles through PP', QQ', and RR'; and the three circles on the three segments PP", QQ', and RR' as diameters are co-axal.

When the order of PQR is opposite that of P'Q'R' as in the figure, and the centre 0 lies outside the points, the co-axal circles are of the l.p.-species, and when the two triads of points have the same order, the co-axal circles are of the c.p.-species.

Cor. 2. Considering ABC as a triangle and AD, BD, CD three lines through its-vertices at D, we have--

The three sides of any triangle and three concurrent lines through the vertices cut any transversal in a six-point involution.

EXERCISES.

1. A circle and an inscribed quadrangle cut any line through

them in involution. 2. The circles of a co-axal system cut any line through them

in involution. 3. Any three concurrent chords intersect the circle in six

points forming a system in involution. 4. The circles of a co-axal system cut any other circle in

involution. 5. Any four circles through a common point have their six

radical axes forming a pencil in involution.

INDEX OF DEFINITIONS, TERMS, ETC.

The Numbers refer to the Articles.

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

IOI

Angles, Vertical,

39
Sum and Difference of, 35

of the same Affection, 65
Anharmonic Ratio, . 298
Antihomologous,

289
Apothem,

146
Arc,
Area,

136
of a Triangle,

1752
Axiom,

3
Axis of a Range,

230
Perspective, .

254
Similitude,

294
Symmetry,

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]

ܕܕ

1

36
36

[ocr errors]
[ocr errors]

Addition Theorem for Sine
and Cosine, .

236
Altitude, .

87
Ambiguous Case,

66
Angle,

31
Acute,

40
Adjacent internal,

49
Basal,

49
External,

49
Obtuse,

40
Re-entrant,

89
Right,
Straight,
Arms of,

32
Bisectors of, .

43
Complement of 40
Cosine of,
Measure of, 41, 207
Sine of,

213
Supplement of,

40
Tangent of,

213
Vertex of,

32
Angles, Adjacent,

35
Alternate,

73
Interadjacent, 73
Opposite,

39

,

IOI

[ocr errors]

49

213

22

7

Basal Angles,
Biliteral Notation,
Bisectors of an Angle,
Brianchon's Theorem,
Brianchon Point,

43

320

[ocr errors]

320

[ocr errors]
[ocr errors][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small]
[ocr errors]
[ocr errors]

Diameter,

95
Difference of Segments, 29
Dimension,

27
Double Point, .

109, 328

22

[ocr errors]
[ocr errors][ocr errors][ocr errors]

86, 97

[ocr errors]
[ocr errors]

Eidograph,

2013
End-points,
Envelope,

223
Equal,

27, 136
Equilateral Triangle,

53
Excircle, .

131
External Angle,

49
Extreme and Mean Ratio, 183
Extremes,

193

[ocr errors]

21

Centre-line,

95
Centre-locus,

129
Centroid,

85
Circle,

92
Circle of Antisimilitude, 290
Inversion, .

256
Similitude,

288
Circumangle,

36
Circumcentre,
Circumcircle,

97
Circumference,

92
Circumradius, .

97
Circumscribed Figure, 97
Chord,

95
Chord of Contact,

114
Co-axal Circles,

273
Collinear Points, 131, 247
Commensurable,

150
Complement of an Angle, 40
Concentric Circles,

93
Conclusion,

4
Concurrent Lines, 85, 247
Congruent,

51
Concyclic,

97
Conjugate Points, etc., 267
Contact of Circles,

291
Continuity, Principle of, . 104
Corollary,

8
C.-P. Circles,

274
Cosine of an Angle,

213
Constructive Geometry, 117
Curve,

15

Finite Line,
Finite Point,

[merged small][ocr errors]

Generating Point,
Geometric Mean,
Given Point and Line,

69
169

[merged small][ocr errors]

Harmonic Division,

208
Harmonic Ratio,

299
Harmonic Systems, 313, 314
Homogeneity, .

160
Homographic Systems, 327
Homologous Sides, . 196
Homologous Lines and
Points,

196
Hypothenuse,

88, 168
Hypothesis,

4.

[ocr errors]
[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors]
[ocr errors]
[ocr errors]

2II

[ocr errors]

36

[ocr errors]

I2

[ocr errors]

21

[ocr errors]

12

[ocr errors]

102

IO

Interadjacent Angles,

Orthogonal Projection, 167, 229

73
Inverse Points,

179, 256
Inverse Figures,

260
Pantagraph,

211)
Involution,

332
Parallel Lines,

70
Isosceles Triangle,

53
Parallelogram,

80
Pascal's Hexagram,

319
Join,

167
Pascal Line,

319

Peaucellier's Cell,
Limit,

148
Pencil,

203
Limiting Points,

274

Perigon,
Line,

Perimeter,

146
Line in Opposite Senses, . 156

Perspective,

254
Line-segment,

Perspective, Axis of,

254
Locus,

69

Centre of, 254
L.-P. Circles,

274
Perpendicular, .

40

Physical Line, .
Magnitude,

190
Plane,

IO, 17
Major and Minor,

Plane Geometric Figure,
Maximum,

175

Plane Geometry,
Mean Centre,

238
Point,

13
Mean Proportional. .

169
Point of Bisection,

30
Means,

193
Contact,

109
Measure,

150

Point, Double,
Median,

55

Pole,
Median Section,

183
Polar,

266
Metrical Geometry,

150
Polar Reciprocal,

268
Minimum,

175
Polar Circle, Centre, etc.,

266
Polygon, .

132

Prime Vector, .
Nine-points Circle,

1265, Ex. 2
Projection,

167
Normal Quadrangle,

89
Proportion,

192

Proportional Compasses, · 2111
Obtuse Angle, .

40
Protractor,

123
Opposite Angles,

39
Opposite Internal Angles, 49
Origin,

1

80, 89
Orthocentre,

88 Quadrilateral,
Orthogonally,

115 Quadrilateral, complete, . 247

II

[ocr errors]
[ocr errors]

109, 328
32, 266

[ocr errors]
[ocr errors]
[merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][ocr errors]
« PreviousContinue »