Editing Butterfly theorem (section)

==Proof==
[[File:Butterfly1.svg|thumb|upright=1.0|{{center|Proof of Butterfly theorem}}]]
A formal proof of the theorem is as follows:
Let the [[perpendiculars]] {{math|''XX′''}}  and {{math|''XX″''}} be dropped from the point {{math|''X''}} on the straight lines {{math|''AM''}} and {{math|''DM''}} respectively. Similarly, let {{math|''YY′''}} and {{math|''YY″''}} be dropped from the point {{math|''Y''}} perpendicular to the straight lines {{math|''BM''}} and {{math|''CM''}} respectively.

Since 
:: <math> \triangle MXX' \sim \triangle MYY',</math>
: <math> {MX \over MY} = {XX' \over YY'}, </math>

:: <math> \triangle MXX'' \sim \triangle MYY'',</math>
: <math> {MX \over MY} = {XX'' \over YY''}, </math>

:: <math> \triangle AXX' \sim \triangle CYY'',</math>
: <math> {XX' \over YY''} = {AX \over CY}, </math>

:: <math> \triangle DXX'' \sim \triangle BYY',</math>
:<math> {XX'' \over YY'} = {DX \over BY}. </math>

From the preceding equations and the [[intersecting chords theorem]], it can be seen that

: <math> \left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''}, </math>

: <math> {} = {AX \cdot DX \over CY \cdot BY}, </math>

: <math> {} = {PX \cdot QX \over PY \cdot QY}, </math>

: <math> {} = {(PM-XM) \cdot (MQ+XM) \over (PM+MY) \cdot (QM-MY)}, </math>

: <math> {} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}, </math>

since {{math|''PM'' {{=}} ''MQ''}}.

So,

:<math> { (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}. </math>

Cross-multiplying in the latter equation,

:<math> {(MX)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} = {(MY)^2 \cdot (PM)^2 - (MX)^2 \cdot (MY)^2} . </math>

Cancelling the common term

:<math> { -(MX)^2 \cdot (MY)^2} </math>

from both sides of the equation yields

:<math> {(MX)^2 \cdot (PM)^2} = {(MY)^2 \cdot (PM)^2}, </math>

hence {{math|''MX'' {{=}} ''MY''}}, since MX, MY, and PM are all positive, real numbers.

Thus, {{math|''M''}} is the midpoint of {{math|''XY''}}.

Other proofs exist,<ref>Martin Celli, "A Proof of the Butterfly Theorem Using the Similarity Factor of the Two Wings", ''[[Forum Geometricorum]]'' 16, 2016, 337–338. http://forumgeom.fau.edu/FG2016volume16/FG201641.pdf</ref> including one using [[projective geometry]].<ref>[http://www.imomath.com/index.php?options=628&lmm=0], problem 8.</ref>