Cross-Ratio Invariance Under Projective Transformations

Represent the four collinear points by homogeneous coordinate vectors $A, B, C, D \in \mathbb{R}^3$ lying in a common 2-dimensional subspace, and represent $\varphi$ by an invertible $3\times 3$ matrix $M$, so that $\varphi$ applied to a point with vector $P$ has homogeneous coordinates $MP$.

Because $A, B, C, D$ are collinear, representatives can be chosen so that $C = \lambda_1 A + \mu_1 B$ and $D = \lambda_2 A + \mu_2 B$ for scalars $\lambda_1, \mu_1, \lambda_2, \mu_2$; the cross-ratio $(A,B;C,D)$ is defined purely in terms of these four scalars, specifically as $\dfrac{\mu_1/\lambda_1}{\mu_2/\lambda_2}$.

Applying the linear map $M$ gives $MC = \lambda_1(MA) + \mu_1(MB)$ and $MD = \lambda_2(MA) + \mu_2(MB)$, because matrix multiplication distributes over linear combinations; so the very same scalars $\lambda_1, \mu_1, \lambda_2, \mu_2$ express $\varphi(C)$ and $\varphi(D)$ in terms of $\varphi(A)$ and $\varphi(B)$.

Since the cross-ratio depends only on these scalars and they are unchanged by $M$, we conclude $(\varphi(A),\varphi(B);\varphi(C),\varphi(D)) = (A,B;C,D)$, which is exactly the claimed invariance.