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.
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.
|What if applet does not run?|
There are several proofs of this statement. The shortest employs the notion of cross-ratio.
(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:
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.
- 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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