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.

Butterfly in Inscriptible Quadrilateral- problem

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.

Butterfly in Inscriptible Quadrilateral- solution

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

  1. Michael Woltermann, Desargues’ Involution Theorem

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
  21. Butterfly Trigonometry
  22. Butterfly in Kite
  23. Butterfly with Menelaus
  24. William Wallace's 1803 Statement of the Butterfly Theorem
  25. Butterfly in Inscriptible Quadrilateral

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