Another synthetic proof of
the butterfly theorem using
the midline in triangle

Tran Quang Hung
Forum Geometricorum, Volume 16 (2016) 345346

Let $M\;$ be the midpoint of a chord $AB\;$ of a circle. Through $M\;$ two other chords $CD\;$ and $EF\;$ are drawn. If $C\;$ and $F\;$ are on opposite sides of $AB,\;$ and $CF,\;$ $DE\;$ intersect $AB\;$ at $G\;$ and $H\;$ respectively, then $M\;$ is also the midpoint of $GH.$

Butterfly by midline - a synthetic proof by Tran Quang Hung

Let $P\;$ be the point on segment $ME\;$ such that $GP\parallel AE.\;$ $PB\;$ intersects $EH\;$ at $Q.\;$ We have

$\angle PGB = \angle EAB = \angle EFB = \angle PFB.$

This shows that quadrilateral $FGPB\;$ is cyclic. We get

$\angle QBM = \angle PBG = \angle PFG = \angle EFC = \angle EDC = \angle QDM.$

Therefore, quadrilateral $DMQB\;$ is also cyclic. From this,

$\angle QMB = \angle QDB = \angle EDB = \angle EAB,$

and $MQ\parallel AE.\;$ Since $M\;$ is the midpoint of $AB,\;$ by the midline theorem, $MQ\;$ passes through the midpoint $R\;$ of $EB.\;$ By Ceva's theorem for $\Delta MEB\;$ and Thales's theorem for $\Delta MEA,\;$ we get $\displaystyle\frac{MH}{MB} = \frac{MP}{ME} = \frac{MG}{MA}.\;$ Since $MA = MB,\;$ we also have $MG = MH.$

Butterfly Theorem and Variants

  1. Butterfly theorem
  2. 2N-Wing Butterfly Theorem
  3. Better Butterfly Theorem
  4. Butterflies in Ellipse
  5. Butterflies in Hyperbola
  6. Butterflies in Quadrilaterals and Elsewhere
  7. Pinning Butterfly on Radical Axes
  8. Shearing Butterflies in Quadrilaterals
  9. The Plain Butterfly Theorem
  10. Two Butterflies Theorem
  11. Two Butterflies Theorem II
  12. Two Butterflies Theorem III
  13. Algebraic proof of the theorem of butterflies in quadrilaterals
  14. William Wallace's Proof of the Butterfly Theorem
  15. Butterfly theorem, a Projective Proof
  16. Areal Butterflies
  17. Butterflies in Similar Co-axial Conics
  18. Butterfly Trigonometry
  19. Butterfly in Kite
  20. Butterfly with Menelaus
  21. William Wallace's 1803 Statement of the Butterfly Theorem
  22. Butterfly in Inscriptible Quadrilateral
  23. Camouflaged Butterfly
  24. General Butterfly in Pictures
  25. Butterfly via Ceva
  26. Butterfly via the Scale Factor of the Wings
  27. Butterfly by Midline
  28. Stathis Koutras' Butterfly
  29. The Lepidoptera of the Circles
  30. The Lepidoptera of the Quadrilateral
  31. The Lepidoptera of the Quadrilateral II
  32. The Lepidoptera of the Triangle
  33. Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
  34. Two Butterfly Theorems by Sidney Kung

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