Butterfly theorem

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For the "butterfly lemma" of group theory, see Zassenhaus lemma.
Butterfly theorem.svg

The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:

Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.

A formal proof of the theorem is as follows: Let the perpendiculars XX'\, and XX''\, be dropped from the point X\, on the straight lines AM\, and DM\, respectively. Similarly, let YY'\, and YY''\, be dropped from the point Y\, perpendicular to the straight lines BM\, and CM\, respectively.

Proof of Butterfly theorem

Now, since

 \triangle MXX' \sim \triangle MYY',\,
 {MX \over MY} = {XX' \over YY'},
 \triangle MXX'' \sim \triangle MYY'',\,
 {MX \over MY} = {XX'' \over YY''},
 \triangle AXX' \sim \triangle CYY'',\,
 {XX' \over YY''} = {AX \over CY},
 \triangle DXX'' \sim \triangle BYY',\,
 {XX'' \over YY'} = {DX \over BY},

From the preceding equations, it can be easily seen that

 \left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''},
 {} = {AX.DX \over CY.BY},
 {} = {PX.QX \over PY.QY},
 {} = {(PM-XM).(MQ+XM) \over (PM+MY).(QM-MY)},
 {} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2},

since PM \, = MQ \,


 { (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}.

So, it can be concluded that MX = MY, \, or M \, is the midpoint of XY. \,

An alternate proof using projective geometry can be found in problem 8 of the link below.



H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967.

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