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This is problem 2, chapter 2 from CAGD 5th edition by Gerald Farin.

In order to prove that any ratio-preserving map $Φ$ is affine I tried to prove that it leaves barycentric combinations invariant.

That is, if $x=\sum a_i*x_i$ with $\sum ai=1$ where $x,x_i \in E^3,a_i \in R$, then $Φx=\sum (a_i*Φx_i)$

To do this I first proved that for any three collinear points $a,x,c$ with $x$ having barycentric coordinates $(A,C)$, $Φx$ also has barycentric coordinates $(A,C)$ with respect to $Φa,Φc$

However, I am having trouble proceeding with the exercise. One of my problems is that in any barycentric combination, the points involved are not always collinear, rendering me unable to use the same method as my first proof.

Any hints or complete answers would be very much appreciated.

Thanks in advance

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