# Another synthetic proof of

the butterfly theorem using

the midline in triangle

### Tran Quang Hung

Forum Geometricorum, Volume 16 (2016) 345–346

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 $GH.$

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

- 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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