Mutually Polar Triangles

The main purpose of this document is to prove the following facts using barycentric coordinates:

Proposition 1. Given a reference triangle and a conic, there exists a triangle $A'B'C'$ such that $ABC$ and $A'B'C'$ are perspective and  mutua\-lly polar with respect to the conic. 

Proposition 2. Let $ABC$ be a triangle and $P$ a point. If $A'B'C'$ is a triangle with perspector $P$ there exists a conic such that $ABC$ and $A'B'C'$ are mutually polar with respect to the conic. 

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