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

[blocks in formation]

.. 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',

¿CPC'=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}

P

D

C

R

A

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

T

R'

B

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(302°)

..

{PQRR'} = {P'Q'R'R},

and the six points are in involution.

Cor. I. 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.

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