The properties of the conic through five points are derived using only simple geometry.
This includes: -point on the conic -tangent at a point -type of conic (ellipse, parabola, hyperbola) -center of the conic -asymptotes (hyperbola) -axes -focal points
It uses the classical projective geometry approach and Pascal's theorem but also a different (and IMHO more elegant) approach shown in steps 4-5 of the construction.