# Areal Butterflies

### Sidney Kung

May 14, 2012

Circles \(O_1\) and \(O_2\) intersect in points \(M\) and \(N\). Line passing through \(M\) intersect\(O_1\) and \(O_2\) in \(A\) and \(B\), respectively. Line passing through \(N\) intersect\(O_1\) and \(O_2\) in \(C\) and \(D\), respectively. If \(AB\) does not intersect \(CD\), and if \(AD\cap BC=I\), then \([\Delta AIB]=[\Delta CID]\), where, \([\Omega ]\) indicates the area of figure \(\Omega\).

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Copyright © 1996-2017 Alexander BogomolnyCircles \(O_1\) and \(O_2\) intersect in points \(M\) and \(N\). Line passing through \(M\) intersect\(O_1\) and \(O_2\) in \(A\) and \(B\), respectively. Line passing through \(N\) intersect\(O_1\) and \(O_2\) in \(C\) and \(D\), respectively. If \(AB\) does not intersect \(CD\), and if \(AD\cap BC=I\), then \([\Delta AIB]=[\Delta CID]\), where, \([\Omega ]\) indicates the area of figure \(\Omega\).

### Proof

There are two cases to consider:

\(A,M,N,C\in O_{1},\space \alpha +\gamma = \pi\) \(B,M,N,D \in O_{2},\space \gamma = \beta\) Therefore, \(\alpha + \beta = \pi\) \(\Rightarrow \space AC||BD\) \(\Rightarrow \space [\Delta ACB]=[\Delta CAD]\) \(\Rightarrow \space [\Delta AIB]=[\Delta CID]\). | ||

\(A,M,N,C\in O_{1},\space \alpha = \gamma\) \(B,M,N,D \in O_{2},\space \gamma = \beta\) Therefore, \(\alpha = \beta\) \(\Rightarrow \space AC||BD\) \(\Rightarrow \space [\Delta BDA]=[\Delta BDC]\) \(\Rightarrow \space [\Delta AIB]=[\Delta CID]\). |

### 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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