Chasles' Theorem, a Proof


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Chasles' Theorem

Let A, B, C, D be four distinct points on a proper conic and let the tangents to the conic at A, B, C, D meet a fixed tangent t to the conic in the points A', B', C', D' respectively. Then, if K is any point on the conic, K(ABCD) = (A'B'C'D').

Proof

Since the theorem is a projective one, it suffices to establish it for the case where the proper conic is a circle. Let O be the center of the circle, and T the point of contact of the tangent t. Then

(1) ∠A'OT = ½∠AOT = ∠AKT.

It follows that pencils K(ABCD) and O(A'B'C'D') are congruent, and

K(ABCD) = O(A'B'C'D') = (A'B'C'D').

Note

  1. The proof is taken verbatim from [Eves, p. 256]. In this form the proof is incomplete as (1) does not always hold. Please see what modifications are needed as K travels over the circle.

  2. As a consequence of the theorem, we have a statement (also referred to as Chasles' theorem) that asserts that the cross-ratio K(ABCD) is independent of K.

References

  1. H. Eves, A Survey of Geometry, Allyn and Bacon, 1972

Chasles' Theorem

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