Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular

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Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$.

Show that it is nonsingular

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Since no three of the five points are aligned, any conic through the five points is non-singular.