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An affine invariant for 4 coplanar points $A,B,C,D$ is: $\mathrm{Area}(ACD)/\mathrm{Area}(ABC)$.

Can somebody provide a proof that this is invariant under affine transformations?

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The ratio of areas of $ABC$ and $ACD$ is the ratio in which the line $AC$ divides the segment $BD$ (and it is the ratio of the heights of $B$ and $D$ over $AC$ respectively). This later ratio is affine invariant as affine transformations preserve length ratios on any line. Do make sure that your points don't collapse onto a single line though.

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