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Classical Geometry: Euclidean, Transformational, Inversive, and Projective by G. W. Tokarsky, A. C. F. Liu, J. E. Lewis, I. E. Leonard

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CHAPTER 14

RECIPROCATION AND THE EXTENDED PLANE

14.1 Harmonic Conjugates

If A and B are two points on a line, any pair of points C and D on the line for which

equation

is said to divide AB harmonically. The points C and D are then said to be harmonic conjugates with respect to A and B.

Lemma 14.1.1. Given ordinary points A and B, and given a positive integer k where k ≠ 1, there are two ordinary points C and D such that

equation

One of the points C and D is between A and B, while the other is exterior to the segment AB.

Proof. Choose a point C on the line AB such that

equation

Since k > 0, then CB < AB, and we may assume that C lies between A and B.

Now, we have

equation

so that

equation

that is,

equation

Now we find the point D, which will be exterior to the segment AB—beyond B if k > 1 and beyond A if 0 < k < 1.

Assuming that k > 1, ...

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