Archive for March, 2015

The Butterfly Theorem

March 1, 2015

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 $XY$. 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).