# 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

Let L be a nondegenerated second degree curve. Suppose, three lines in the plane of L concur at the point M. Let points A, B; P, Q; and C, D be the intersections of these lines with L, and let AD and BC intersect PQ at X and Y, respectively. Finally, let N be the point harmonic conjugate to M with respect to the pair of points

(P, Q; M, N) = -1 ⇒ (X, Y; M, N) = -1.

### 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

This completes the proof.

### Butterfly Theorem and Variants

- Butterfly theorem
- 2N-Wing Butterfly Theorem
- Better Butterfly Theorem
- Butterflies in Ellipse
- Butterflies in Hyperbola
- Butterflies in Quadrilaterals and Elsewhere
- Pinning Butterfly on Radical Axes
- Shearing Butterflies in Quadrilaterals
- The Plain Butterfly Theorem
- Two Butterflies Theorem
- Two Butterflies Theorem II
- Two Butterflies Theorem III
- Algebraic proof of the theorem of butterflies in quadrilaterals
- William Wallace's Proof of the Butterfly Theorem
- Butterfly theorem, a Projective Proof
- Areal Butterflies
- Butterflies in Similar Co-axial Conics
- Butterfly Trigonometry
- Butterfly in Kite
- Butterfly with Menelaus
- William Wallace's 1803 Statement of the Butterfly Theorem
- Butterfly in Inscriptible Quadrilateral
- Camouflaged Butterfly
- General Butterfly in Pictures
- Butterfly via Ceva
- Butterfly via the Scale Factor of the Wings
- Butterfly by Midline
- Stathis Koutras' Butterfly
- The Lepidoptera of the Circles
- The Lepidoptera of the Quadrilateral
- The Lepidoptera of the Quadrilateral II
- The Lepidoptera of the Triangle
- Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
- Two Butterfly Theorems by Sidney Kung

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