(AB, C′A′) = (A′B′, CA),
for each expresses that in the two projective rows in which A, B, C and A′, B′, C′ are conjugate points two conjugate elements may be interchanged.
8. Any three pairs. A, A′, B, B′, C, C′, of conjugate points are connected by the relations:
| AB′ · BC′ · CA′ | = | AB′ · BC · C′A′ | = | AB · B′C′ · CA′ | = | AB · B′C · C′A′ | = −1. |
| A′B · B′C · C′A | A′B · B′C′ · CA | A′B′ · BC · C′A | A′B′ · BC′ · CA |
These relations readily follow by working out the relations in (7) (above).
§ 78. Involution of a quadrangle.—The sides of any four-point are cut by any line in six points in involution, opposite sides being cut in conjugate points.
Let A1B1C1D1 (fig. 31) be the four-point. If its sides be cut by the line p in the points A, A′, B, B′, C, C′, if further, C1D1 cuts the line A1B1 in C2, and if we project the row A1B1C2C to p once from D1 and once from C1, we get (A′B′, C′C) = (BA, C′C).
Interchanging in the last cross-ratio the letters in each pair we get (A′B′, C′C) = (AB, CC′). Hence by § 77 (7) the points are in involution.
The theorem may also be stated thus:
The three points in which any line cuts the sides of a triangle and the projections, from any point in the plane, of the vertices of the triangle on to the same line are six points in involution.