is there any thing missing.

Not as far as Imomath tells me. I checked with a diagram too.

if the concurrency is met then $ABCDEF$ should be inscriptible in a conic but look at the example in the diagram below

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@above it says $\overline{AC}\cap\overline{DF}$. By Pascal's the hexagon's vertices are coconic. Affine transform their circumconic to a circle. By Desargues we have to show that $\overline{AD},\overline{BE},\overline{CF}$ are concurrent. Since they all pass through the center of the circle, we are done.

Thanks for indicating my misreading.