# Proof that a Steiner's conic is a projective conic

In class, I've defined a projective conic on $$\mathbb{P}(V)$$ as an element of $$\mathbb{P}(Q(V))$$, where $$V$$ is a $$K$$-vector space and $$Q(V)$$ is the field of quadratic forms on $$V$$. I need to prove the following:

Given two points $$A,B$$ of the complex or real projective plane, the pencils $$\mathcal{H}_A,\mathcal{H}_B$$ of lines at $$A$$ and $$B$$ respectively, and a projectivity $$\psi:\mathcal{H}_A\to\mathcal{H}_B$$, the set of points $$r\ \cap \psi (r)$$ are all contained in a conic.

So, what I need is to prove that both definitions are equivalent (for a $$2$$-dimensional projective space).

EDIT: I'm thinking of using the immersion of the affine into the projective plane, and hopefully be able to define an affine conic that will give me the projective conic I am looking for.