The Lepidoptera of the Quadrilateral II
Sidney Kung
April 26, 2008
We prove the Butterfly Theorem with the aid of Menelaus' theorem.
Through the intersection I of the diagonals AC, BD of a convex quadrilateral ABCD, draw two lines EF and HG that meet the sides of ABCD in E, F, G, H. Let M and N be the intersections of EG and FH with AC. Then
1/IM  1/IA = 1/IN  1/IC.
Proof
Figure 1. 
Refer to Figure 1. Triangles BAD and CAD are cut by transversal EIF. So, by applying Menelaus' theorem twice, we have
BE/EA · AK/KD · DI/IB = 1,
CF/FD · DK/KA · AI/IC = 1.
Multiplying the above gives
BE/EA · ID/IB · CF/FD · IA/IC = 1,
or
(1)  IA · ID · BE/EA = IC · IB · FD/CF. 
Similarly, since transversal HIG cuts triangles BDC and ADC, applying Menelaus' theorem twice again will give us
AG/GD · DI/IB · BH/HC · CI/IA = 1,
or
(2)  ID · IC · BH/HC = IB · IA · DG/GA. 
Note that neither (1) nor (2) involves IM, MA, IN, or NC. The relationship among these and other line segments in the figure will be established as follows:
Figure 2. 
Refer to Figure 2. Let L and P be the intersections of FH and EG with BG. Consider the triangle CDI with transversal FH, we have
DF/ FC · CN/NI · IL/LD =1,
or
(3)  DF/FC = IN/NC · LD/IL. 
Similarly, from triangle CIB, we see that
BH/HC · CN/NI · IL/LB = 1,
or
(4)  BH/HC = NI/CN · LB/IL. 
Thus,
(5) 

Again, we apply the Menelaus theorem on triangles AIB and ADI to get
(6)  BE/EA = MI / AM · PB/IP 
and
(7)  DG / GA = MI / AM · PD/IP, 
respectively. So,
(8) 

Now, multiplying (5) by IC, we get
(9)  IC · IB · DF/FC + IC · ID · BH/HC = IC · BD · IN/NC 
and multiply (8) by IA, we have
(10)  IA · IB · DG/GA + IA · ID · BE/EA = IA · BD · IM/MA. 
Finally, by subtracting (10) from (9), and taking into account (1) and (2), we obtain
BD · ((IC · IN)/NC  (IA · IM)/MA ) = 0.
It follows then that
(IA  IM)/(IA · IM) = (IC  IN) / (IC · IN).
Therefore, 1/IM  1/IA = 1/IN  1/IC, as required
Butterfly Theorem and Variants
 Butterfly theorem
 2NWing 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 Coaxial 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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