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Theorem:
Any section of a harmonic set of lines is a harmonic set of points, and a harmonic set of points is projected from any point by a harmonic set of lines.
Remark: This theorem is in two dual parts, and thus it will suffice to prove the latter part: If A,B,C,D are joined to a point P (outside their line) by lines a,b,c,d and if H(AB,CD), then H(ab,cd).
Proof:
Let P be used as a vertex of the triangle PQR in constructing D from A,B,C. Then the quadrilateral AD,DS,SQ,QA has two opposite vertices on AS=a, two others on BR=b, one vertex Q on c, and one vertex D on d. Hence H(ab,cd)
Combining the above, we conclude that
Theorem:
Perspectivities preserve the harmonic relation:
If ABCD =/\= A'B'C'D' and H(AB,CD), then H(A'B',C'D')
Our definition for harmonic conjugacy involves A and B symmetrically and likewise C and D. Hence
Theorem: H(AB,CD), H(AB,DC), H(BA,DC), H(BA,CD) are all equivalent.e their line) by lines a,b,c,d and if H(AB,CD), then H(ab,cd).
Proof:
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