Harmonic Ratio

Given three collinear points A, B, and C. Perform the following construction:

Choose point E not collinear with A and B.
Connect E to A, B, and C.
On AE choose point I and connect it to B.
Let G be the intersection of CE and BI. Extend line AG to its intersection with BE at F.
Extend FI to its intersection with AB at D.

Theorem

D does not depend on the choice of E and I, but only on A, B, and C.

In other words, for the given A, B, and C, the above construction always leads to the same point D.

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There are several proofs of this statement. The shortest employs the notion of cross-ratio.

First, obviously,

(ABCD) = E(ABCD) = E(IFHD) = (IFHD).

As obvious are also the following identities,

(IFHD) = G(IFHD) = G(BACD) = (BACD).

From the definition (ABCD) = CA/CB : DA/DB; thus it follows that

(ABCD) = 1/(BACD).

Combining the above, (ABCD)² = 1, i.e., (ABCD) = ±1. Considering the relative positions of the four points A, B, C, D, (ABCD) is bound to be negative: (ABCD) = -1. The latter identity uniquely identifies point D.

Point D is said to be harmonic conjugate of C with respect to the pair A, B. Clearly, if D is the harmonic conjugate of C, then C is the harmonic conjugate of D. Thus the pair C, D is often said to be (harmonically) conjugate with the pair A, B, and vice versa.

Note that H is the conjugate of D with respect to the pair I, F, and that the line CH passes through E. This leads to another important fact (and a definition.)

Given angle E. For any point D draw a secant that cuts the lines forming angle E at points A and B. The locus of points C conjugate to D with respect to A and B is a straight line through E. The latter is called the polar of D with respect to the angle E (or two lines AE, BE, or the corresponding degenerate conic section.) D is called the pole of EC.

The above theorem corroborates a straight edge only construction of harmonic conjugates and polars. Besides the line AB itself, it takes 6 more lines to obtain point C. Of these, four lines (drawn in black in the applet) form a complete quadrilateral, for which the remaining lines (drawn in blue) serve as diagonals. The harmonic conjugates of D and D itself lie at the intersection of the diagonals. The theorem then admits the following interpretation:

Harmonic Ratio in Complete Quadrilateral

Diagonals of complete quadrilateral meet each other in three points each pair of which is conjugate to a pair of vertices. For example, C and D is a pair conjugate with respect to A and B, while C, H are conjugate with respect to E, G, and finally H, D are conjugate with respect to I, F.

References

R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1996
D. Wells, The Penguin Dictionary of Curious and Intersting Geometry, Penguin Books, 1991

Poles and Polars

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