Butterfly theorem, a Projective Proof

The Butterfly Theorem is an obvious corollary of the following statement. This observation and the proof that follows are due to Mikhail Goldenberg (The Ingenuity Project, Baltimore, MD) and Mark Kaplan (Towson State University)

Theorem

Proof

First, note that if a rectangle can be inscribed in a nondegenerated second degree curve, then the curve is a central one and its center coincides with the center of the rectangle. Second, the projective transformation preserves the cross ratio. Therefore, it is sufficient to prove the theorem for some projective image of the given configuration.

There exists a projective transformation of the plane of L which maps the quadrilateral ACBD onto a square A'C'B'D'. The image M' of the point M is the center of the square which is, as observed, the center of the image L' of the curve L. Therefore, M' is the midpoint of both segments, P'Q' and X'Y' , and the point N' is at infinity on the line P'Q' because (P, Q; M, N) = -1. So, N' is the harmonic conjugate point for M' with respect to (X', Y).

This completes the proof.

Butterfly Theorem and Variants

1. Butterfly theorem
2. 2N-Wing Butterfly Theorem
3. Better Butterfly Theorem
4. The Lepidoptera of the Circles
5. The Lepidoptera of the Quadrilateral
6. The Lepidoptera of the Quadrilateral II
7. Butterflies in Ellipse
8. Butterflies in Hyperbola
9. Butterflies in Quadrilaterals and Elsewhere
10. Pinning Butterfly on Radical Axes
11. Shearing Butterflies in Quadrilaterals
12. The Plain Butterfly Theorem
13. Two Butterflies Theorem
14. Two Butterflies Theorem II
15. Two Butterflies Theorem III
16. Algebraic proof of the theorem of butterflies in quadrilaterals
17. William Wallace's Proof of the Butterfly Theorem
18. Butterfly theorem, a Projective Proof
19. Areal Butterflies
20. Butterflies in Similar Co-axial Conics

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