# “Special case” of Brianchon's theorem for any conic section.

Let $c$ be an arbitrary conic section. Choose $6$ distinct points on $c$. Draw $6$ lines $t_1,t_2,...t_6$ through these points that are tangents to $c$. Denote points $T_1=t_1\cap t_2,T_2=t_2\cap t_3\cdots T_6=t_6\cap t_1$. Then lines $\leftrightarrow T_1T_4$,$\leftrightarrow T_2 T_5$,$\leftrightarrow T_3,T_6$ meet at 1 point.

I would like to get a reasonable explanation on how this can work or a rigorous proof. I couldn't find a starting point for this. What I am familiar with is the proof of the classic Brianchon's theorem for hexagon circumscribed around a circle, using radix axes, it's easy to prove, but I am struggling with proving this for general conic section. Thanks for any tips and hints.

• There are two topics important to constructing a proof polarity and duality. (On a conic polar and dual are somewhat 'interchangeable') A few things are interesting about the polar/dual of Brianchon's theorem. – Patrick Abraham May 3 '18 at 14:33