We introduce the concept of inversion with a simple example, that of constructing the midpoint of a line segment using only a compass.

**Example 13.1.1.** *Given a line through A and B, find the midpoint of the segment AB using only a compass*.

*Solution*. With center *A* and radius *r* = *AB*, draw the circle *C*(*A, r*) and locate the point *P* on the line *AB* so that *B* is the midpoint of *AP*.

With center *P*, draw the circle *C*(*P, AP*) intersecting the first circle at *C*, as in the figure on the following page.

Finally, draw *C*(*C, r*) intersecting the line *AB* at *P*′. Then *P*′ is the midpoint of *AB*.

To sec that this is the case, note that the triangles *AP′C* and *ACP* are similar isosceles triangles since they share a vertex angle at *A*, so that

which implies that

and this implies that

Note that with

we have *AP · AP*′ = *r*^{2}. This relationship between *P* and *P*′ is called an *inversion*. More generally, we have the following definition: ...

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