__My mother’s proof of the Butterfly Theorem__ (see EWD866)

In a circle with centre `O` the chords `AD` and `BC` intersect one another in `Q`. The line `PQR` intersects `AB` in `P` and `CD` in `R` and is orthogonal to `OQ`. Prove `PQ = QR`.

(The name “Butterfly Theorem” is derived from the shape of the quadrilateral `ABCD`.)

Mother only used that the circle is a conic section and that `Q` divides the dotted chord into two equal halves. With `O` in the origin and the dotted chord of length `2⋅q` along the y-axis, the substitution `x ≔ 0` reduces the equation of the conic section to

y^{2} = q^{2} (0)

The substitution `x ≔ 0` reduces the equation of the (degenerate) conic section consisting of the lines `AD` and `BC` to

y^{2} = 0 (1)

The substitution `x ≔ 0` reduces the equation of any other member of the bundle on those two, i.e. of any other conic section through `A`, `B`, `C`, and `D`, to

y^{2} + λ ⋅ y^{2} = q^{2} + λ ⋅ O or

(1 + λ) ⋅ y^{2} = q^{2} (2)

for some λ. This holds in particular for the (degenerate) conic section consisting of the lines `AB` and `CD`. Since the two roots of (2) have equal absolute values, this concludes the proof.

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Mother’s proof is far superior to mine. I worked with line bundles. By recognizing pairs of lines as conic sections she did it with a bundle of conic sections. And, in passing, she generalized the theorem. To quote Ross Honsberger “Bless her heart!”.

Plataanstraat 5 5671 AL NUENEN The Netherlands |
18 December 1983 prof.dr.Edsger W. Dijkstra Burroughs Research Fellow |