# Butterfly in Inscriptible Quadrilateral

Here's a problem by Dao Thanh Oai with a solution by Telv Cohl.

Let $ABC$ be an inscriptible quadrilateral with the incircle $(O),$ $E=AC \cap BD.$ Let a line through $E$ meets $AD,$ $BC$ in $F$ and $G,$ respectively. Assume also, it meets $(O)$ at $H$ and $I,$ as in the diagram.

Prove that $EI=EH$ iff $EF=EG.$

### Proof

If $(O)$ touches $AD,$ $BC$ at $X,$ $Y,$ respectively, then, as is well known, $X,$ $E,$ $Y$ are collinear. Desargues' Involution Theorem applied to the degenerate quadrangle $XXYY,$ informs us that $E$ is a double fixed point of the involution defined on the given line, with $F,G$ and $H,I$ reciprocal pairs.

The condition $EI=EH$ means that the involution is a symmetry in $E,$ implying $EF=EG.$

**Note**: the proof above is reminiscent of Hubert Shutrick's proof (Proof #20) of the common Butterfly Theorem.

### Reference

- Michael Woltermann, Desargues’ Involution Theorem

### 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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