# Butterfly with Menelaus

### Sidney Kung

September 7, 2012

We give a proof of the following by using Menelaus' theorem:

Through a point \(P\) of the line segment \(AB\) whose end points \(A\) and \(B\) lie on two intersecting lines \(l\) and \(l'\), respectively, draw \(CD\) and \(EF\) \((C, F\in l\), and \(E,D\in l')\). Let \(CE\cap AB=X\), \(FD\cap AB=Y.\space\)Then \(PA = PB\) implies \(PX = PY.\)

### Proof

For a repeated application of Menelaus' here, we shall proceed by focusing on parts of the diagram:

- \(\triangle OFE\) cut by transversal \(CPD\)
\(\frac{OC}{CF}\times \frac{FP}{PE}\times \frac{ED}{DO} = 1\).

- \(\triangle OAB\) cut by transversal \(CPD\)
\(\frac{OD}{DB}\times \frac{BP}{PA}\times \frac{AC}{CO} = 1\).

- \(\triangle PBE\) cut by transversal \(FYD\)
\(\frac{PY}{YB}\times \frac{BD}{DE}\times \frac{EF}{FP} = 1\).

- \(\triangle FAP\) cut by transversal \(CXE\)
\(\frac{PE}{EF}\times \frac{FC}{CA}\times \frac{AX}{XP} = 1\).

Multiplying the four identities and simplifying gives \(\frac{PY}{YB}\times \frac{AX}{XP}\times \frac{BP}{PA} = 1\). Since \(PB=PA\), the latter expression further simplifies to \(\frac{AX}{XP}=\frac{YB}{PY}\). Add now \(1\) to both sides: \(\frac{AX+XP}{XP}=\frac{PY+YB}{PY}\), or \(\frac{PA}{XP}=\frac{PB}{PY},\space\)implying \(PX=PY\).

The above proof moved Hubert Shutrick to make the following observation:

The conics of the pencil through \(C\), \(D\), \(E\), \(F\) intersect the line \(AB\) in pairs of points in **involution**. Consider the **degenerate members** \(CD.EF\), \(CE.DF\), and \(l.l'\). \(P\) is a **double point** because of the first and so, if one of the pairs \(A,B\) and \(X,Y\) is symmetrical about \(P\), then so is the other.

### 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
- Butterfly in Complex Numbers

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