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TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28
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The cross-ratio is invariant under Möbius transformations

Statement

If M is a Möbius transformation and z1,z2,z3,z4 are distinct sphere points, then [M(z1),M(z2);M(z3),M(z4)]=[z1,z2;z3,z4]. Thus the cross-ratio is a Möbius invariant.

Facts & Assumptions

Given: A Möbius transformation M and distinct points z1,z2,z3,z4∈C^.

[L1]

There is a unique Möbius transformation sending any ordered triple of distinct sphere points to any other such triple (A unique Möbius transformation carries any ordered triple of distinct sphere points to any other).

Proof

technique · direct
1.1

Let N be the unique Möbius transformation sending (z2,z3,z4) to (1,0,∞), and let NM be the unique Möbius transformation sending (M(z2),M(z3),M(z4)) to (1,0,∞). The defining formulas for the cross-ratio in its first variable give a Möbius map that sends (z2,z3,z4) to (1,0,∞), so uniqueness gives N(z)=[z,z2;z3,z4] for all z. Likewise NM(w)=[w,M(z2);M(z3),M(z4)] for all w. In particular N(z1)=[z1,z2;z3,z4],NM(M(z1))=[M(z1),M(z2);M(z3),M(z4)].

L1given
2.1

The maps N∘M−1 and NM have the same action on the triple (M(z2),M(z3),M(z4)), so [L1] makes them equal. Evaluating at M(z1) yields [M(z1),M(z2);M(z3),M(z4)]=[z1,z2;z3,z4].

L1given∎

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