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If H(AB,CD) and H(AB,CD*) then D=D*
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See Meserve and Izzo:Section 5-1 page127.
This can be stated as: "The harmonic conjugate
[D] of C with respect to A and B is unique."
Proof: Follow around the triangles
using Desargues' Theorem (and its converse).
Discussion: The proof demonstrates that P2P3, R2R3, and AC = m intersect at the same point, D = D*.
Using Desargues theorem (with its dual/converse): P1P3P4 ....R1R3R4 are perspective with m so there is a point O where P1P3P4 ....R1R3R4 with O and so P1R1, P3R3, and P4R4 all pass through O.
Also P1P2P4 ....R1R2R4 are perspective with m while P1R1 and P4R4 pass through O so P1P2P4 ....R1R2R4 with O.
Now P1P2P3 ....R1R2R3 are perspective with O, and since P1P2 meets
R1R2 at A and P1P3 meets R1R3 at B, then P1P2P3 ....R1R2R3 are
perspective with m and so P2P3 must meet R2R3 on m. thus D = D*. EOP