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In the list of recomended problems for my course "Projective Algebraic Geometry" I found following problem:

Prove that two triangles are inscribed in some conic $C_1$ $\iff$ their edges are tangents of another conic $C_2$ .

All geometry is over field $char \neq 2$. Any ideas how it can be proved?

Thanks!

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Let $ABC$ and $A'B'C'$ be the two triangles inscribed in the same conic. Consider the exagon $ABCC'B'A'$: by Pascal's theorem the three points \begin{align} AB&\cap C'B'& BC&\cap B'A'& CC'&\cap A'A& \end{align} lie on a straight line.

Now, let $a=BC$, $b=AC$, $c=AB$ and $a'=B'C'$, $b'=A'C'$, $c'=A'B'$ be the edges of our triangles and consider the exagon whose edges are $a,b,c,a',b',c'$. The three lines containing the three pairs of opposite vertex are \begin{align} (a\cap b)(a'\cap b')&=CC'\\ (b\cap c)(b'\cap c')&=AA'\\ (c\cap a')(c'\cap a)&=(AB\cap B'C')(A'B'\cap BC) \end{align} and they meets at a point. By Brianchon's theorem $a,b,c,a',b',c'$ are tangent to a conic.

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  • $\begingroup$ Can we apply Brianchon's theorem if our conic is singular? $\endgroup$ – Gleb Chili Dec 14 '17 at 13:01
  • $\begingroup$ Yes, Brianchon's theorem holds for also degenerate conic. $\endgroup$ – Fabio Lucchini Jan 11 '18 at 12:56

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