The butterfly theorem

Setup. Put $M=(0,0)$, $P=(-p,0)$, $Q=(p,0)$. The centre is $(0,c)$ and the circle is $x^2+(y-c)^2=p^2+c^2$. For the line through $M$ with unit direction $u=(\cos\theta,\sin\theta)$, the point $tu$ lies on the circle iff
$$t^2-2ct\sin\theta-p^2=0.$$
Its two roots satisfy $t_1+t_2=2c\sin\theta$ and $t_1t_2=-p^2$.

Parametrising. Let chord $AB$ have direction $u=(\cos\alpha,\sin\alpha)$, $A=t_1u$, $B=t_2u$, and chord $CD$ direction $v=(\cos\beta,\sin\beta)$, $C=s_1v$, $D=s_2v$, with $t_1+t_2=2c\sin\alpha$, $s_1+s_2=2c\sin\beta$, $t_1t_2=s_1s_2=-p^2$.

The intersections. For $X=AD\cap Ox$ one computes $x_X=\dfrac{t_1s_2\sin(\alpha-\beta)}{t_1\sin\alpha-s_2\sin\beta}$, and for $Y=BC\cap Ox$, $x_Y=\dfrac{t_2s_1\sin(\alpha-\beta)}{t_2\sin\alpha-s_1\sin\beta}$.

Showing $x_X+x_Y=0$. This reduces to
$$\sin\alpha\;t_1t_2\,(s_1+s_2)-\sin\beta\;s_1s_2\,(t_1+t_2)=0,$$
which is immediate after substituting Vieta's relations. Hence $x_X=-x_Y$, so $M$ is the midpoint of $XY$. $\blacksquare$

Lesson. Parametrise points of a configuration by distance along lines through a base point; the symmetric functions of the roots (Vieta) then do the work.