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

is said to *divide**AB* ** harmonically.** The points

**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*

*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

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

Now, we have

so that

that is,

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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