§ 24: Tangents and points of contact

Proposition: Let J be a non-degenerated conic of points. Then for each point A of J there is exactly one line through A that isn't incident with any other point of J.

Proof: Take besides A four other points of J, say B, C, D and E.
Then J is the conic of points determined by the projectivity φ from A onto B that maps AC to BC, AD to BD, and AE to BE.
Let P be the Pappus point of this projectivity. Each line x through A has with J the second point x . φ(x) in common.
If x = AP then φ(x) = AB and x . φ(x) = A. So AP is a line as required.
If xAP then φ(x) ≠ AP (for φ is 1-1), so then φ(x) doesn't go through A (for φ(x) does go through B), and then x . φ(x) ≠ A.
So AP is the only line that meets the requirements.

Definition: We call the line through A on J that isn't incident with any other point of J tangent to J in A.

Note: The proof of the last proposition gives a procedure for the construction of the tangent to J in A whenever five points of J are given among which A.

Note: In the situation of the proof of the last proposition, BP is the tangent to J in B. Prove this.

Proposition: Let j be a non-degenerated conic of lines. Then for each line a of j there is exactly one point on a that isn't incident with any other line of j.

Proof: Dualise the proof of the previous proposition.

Definition: We call the point on the line a of j that isn't incident with any other line of j the point of contact to j on a.


O88 Given five freely situated points A, B, C, D, E. Construct the tangent in A to the conic through these five points.

O89 Given five freely situated lines a, b, c, d, e. Construct the point of contact on a to the conic of lines determined by these five lines.

O90 Given three lines a, b, c of a conic of lines j, and the contact points A on a and B on b. Construct more lines on j and the contact point on c.

O91 Given three points A, B, C of a conic of points J, and the tangents a in A and b in B. Construct more points of J and the tangent in C.

Note: It's even easier to solve the problems of this section after studying §26. See O101.