The Lepidoptera of the Triangle

Sidney Kung
July, 2107

Theorem 1

Proof of Theorem 1

We begin by assuming a more general case than that shown in Figure 1, in that

1. none of the three transversals $DE\,$ $FG,\,$ and $HJ\,$ needs to be parallel to a side of $\Delta ABC,$
2. we only ask that the points $P=AB\cap FG,\,$ $Q=HJ\cap BC,\,$ $R=DE\cap AC\,$ be collinear.

Denote the six vertices of the hexagon $HDFJEG\,$ in the order $325416.\,$ We see that the points of intersections of the three pairs of the opposite sides $23-56,\,$ $34-61,\,$ $12-45\,$ are, respectively $P,\,$ $Q,\,$ $R.\,$ Thus, by the converse of Pascal's Theorem, if $P,\,$ $Q,\,$ $R\,$ are collinear, the hexagon $HDFJEG\,$ is inscribed in a conic.

Return to Figure 1. Denote the vertices of the hexagon $HDFJEG\,$ in the order $325416.\,$ Since $FG\parallel AB,\,$ $23-56\,$ meet at a point of infinity. Similarly, the same holds for $34-61\,$ and $12-45.\,$ Thus, the points $P,\,$ $Q,\,$ $R,\,$ all being points at infinity, are collinear, hence the hexagon $HDFJEG\,$ is inscribed in a conic. It follows then from Candy's generalized Butterfly Theorem, that

$\displaystyle \frac{1}{HI}-\frac{1}{IJ}=\frac{1}{IX}-\frac{1}{IY},$

which is (1), and (2) is immediate if $I\,$ is the midpoint of $HJ.$

Theorem 2

Proof of Theorem 2

Consider $DR\,$ and the circumscribe circle of $\Delta AIC$ (see Figure 3ii). It's easy to see that $\Delta RIA\sim\Delta RCI.\,$ So,

$\displaystyle \frac{RA}{RI}=\frac{IA}{IC},\;\frac{RC}{RI}=\frac{AC}{IA}.$

Dividing the former by the latter, we get $\displaystyle \frac{RA}{RI}=\frac{IA^2}{IC^2}.\,$ Similarly, since $\Delta HQC\sim\Delta QBI\,$ and $\Delta PIC\sim\Delta PBI,\,$ $\displaystyle \frac{QC}{QB}=\frac{IC^2}{IB^2}\,$ and $\displaystyle \frac{PB}{PA}=\frac{IB^2}{IA^2}.\,$ Thus

$\displaystyle \frac{PB}{PA}\cdot\frac{QC}{QB}\cdot\frac{RA}{RC}=\frac{IB^2}{IA^2}\cdot\frac{IC^2}{IB^2}\cdot\frac{IA^2}{IC^2}=1.$

By the converse of Menelaus' theorem, the three points $P,Q,R\,$ are collinear, implying that the hexagon $HDFJEG\,$ is inscribed in a conic. Consequently, Candy's theorem delivers (3) and (4).

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
35. Butterfly in Complex Numbers

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