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 \(PX = PY.\)

statement of the butterfly theorem in a quadrilateral

Proof

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

  1. \(\triangle OFE\) cut by transversal \(CPD\)

    first step of the proof

    \(\frac{OC}{CF}\times \frac{FP}{PE}\times \frac{ED}{DO} = 1\).

  2. \(\triangle OAB\) cut by transversal \(CPD\)

    second step of the proof

    \(\frac{OD}{DB}\times \frac{BP}{PA}\times \frac{AC}{CO} = 1\).

  3. \(\triangle PBE\) cut by transversal \(FYD\)

    third step of the proof

    \(\frac{PY}{YB}\times \frac{BD}{DE}\times \frac{EF}{FP} = 1\).

  4. \(\triangle FAP\) cut by transversal \(CXE\)

    fourth step of the proof

    \(\frac{PE}{EF}\times \frac{FC}{CA}\times \frac{AX}{XP} = 1\).

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

  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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