The Butterfly Theorem

Let M be the midpoint of the chord PQ of a circle. Let AB and CD be two other chords of the circle passing through M and let X and Y be the intersection points of PQ with AD and BC respectively. Then M is the midpoint of the segment XY. Butterfly Theorem diagram (from wikipedia) Proof: The space of all degree two polynomials vanishing at the four points A, B, C and D is of dimension 2*. Thus it is spanned by the equation of the circle and the equation of the union of the lines AB and CD. Choose coordinates such that the line PQ is the x-axis and the point M is the origin. Then the coefficient of x in both our spanning polynomials is equal to zero, hence this coefficient is also zero in the equation of the union of AD and BC. Therefore M is the midpoint of the segment XY, as required.

*To see this there is a six dimensional space of degree two polynomials and we are imposing one linear condition each time we require the polynomial to vanish at a point. These four conditions are linearly independent since it is easy to find degree two curves passing through three of these points and not the fourth (e.g. a union of two lines).

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