Synthetic Construction of Harmonic Quadruples

Interactive Course on Projective Geometry

Definition 0.1   A quadruple of points on the projective line is said to be a harmonic quadruple if its cross-ratio equals $ -1$.

Theorem 0.1   In the following figure the quadruple $ x,y,z,t$ is harmonic.

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Exercise 0.1 (05)   Find in the figure two perspectives such that their composition interchanges $ x$ and $ y$ while fixing both $ z$ and $ t$.

Exercise 0.2 (05)   Use that perspectives preserve cross-ratios to deduce the theorem from the previous exercise.