Affine construction of conic sections from their five points (Q6617254)

This paper is devoted to an elementary construction of a conic section passing through five given points in general position. No use is made of Pascal's theorem, nor of any notion from projective geometry. The construction proceeds inside affine geometry and treats the ellipse, hyperbola, and parabola case separately. The construction is based on the fact that the conic section is the following locus of Apollonius: Given four lines $a, b, c, d$ in general position, the locus of all points $P$ in the plane for which the ratio $\frac{d(P, a)d(P, c)}{d(P, b)d(P, d)}$ is fixed forms a conic section.