Page images
PDF
EPUB

333°. Let A, A', B, B', C, C' be six points in involution, and

let O be the centre.

Draw any line OPQ through O,

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, . OA. OA'=OP.OQ,

A B C

B'

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

Similarly,

LAPB=LA'QB'.

LBPC=LB'QC',

LCPC'=4C'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.

334°. 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'} = D{CARB}
= B{CARD}

=B{Q'P'RR'} = {P'Q'R'R},

T

B

(302°)

..

{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 O 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.

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