Pencil of Conics
§ 87. Through four points A, B, C, D in a plane, of which no three lie in a line, an infinite number of conics may be drawn, viz. through these four points and any fifth one single conic. This system of conics is called a pencil of conics. Similarly, all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic. We have here the theorems:
| The pairs of points in which any line is cut by a system of conics through four fixed points are in involution. | The pairs of tangents which can be drawn from a point to a system of conics touching four fixed lines are in involution. |
| Fig. 36. |
We prove the first theorem only. Let ABCD (fig. 36) be the four-point, then any line t will cut two opposite sides AC, BD in the points E, E′, the pair AD, BC in points F, F′, and any conic of the system in M, N, and we have A(CD, MN) = B(CD, MN).
If we cut these pencils by t we get
(EF, MN) = (F′E′, MN)
or
(EF, MN) = (E′F′, NM).
But this is, according to § 77 (7), the condition that M, N are corresponding points in the involution determined by the point pairs E, E′, F, F′ in which the line t cuts pairs of opposite sides of the four-point ABCD. This involution is independent of the particular conic chosen.