Polarit\'es d\'efinies par un triangle

A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebraic curve, the polarities with respect to a convex set. In this article, we introduce a polarity with respect to a triangle motivated by a question on duality of projective frames. We show that the four polarities above apply to a triangle and in fact coincide. This result is an opportunity to review nice concepts of projective geometry, linear algebra and convex geometry.