Carnot's theorem (conics)

Carnot's theorem (named after Lazare Carnot) describes a relation between conic sections and triangles.

In a triangle $$ABC$$ with points $$C_A, C_B$$ on the side $$AB$$, $$A_B, A_C$$ on the side $$BC$$ and $$B_C, B_A$$ on the side $$AC$$ those six points are located on a common conic section if and only if the following equation holds:



\frac{|AC_A|}{|BC_A|}\cdot \frac{|AC_B|}{|BC_B|}\cdot \frac{|BA_B|}{|CA_B|}\cdot \frac{|BA_C|}{|CA_C|} \cdot \frac{|CB_C|}{|AB_C|}\cdot \frac{|CB_A|}{|AB_A|}=1 $$.