InverseLengths

Inverse Lengths

Let c(O,r) be a circle and AB a segment. Consider segment A'B' resulting by inverting the end-points of AB with respect to (c). Show that the segments lengths are related by:

|A'B'| = (r² * |AB|)/(|OA|*|OB|).

Triangles OAB and OA'B' are similar. Thus, |A'B'|/|AB| = |OA'|/|OB| = (|OA'|*|OA|)/(|OB|*|OA|) = r²/(|OB|*|OA|).
The figure contains some other geometric relations. State and prove them. In particular, show that the ratio of the sides of the parallelogram is also |OC|/|OD| = r²/(|OB|*|OA|).
Notice that if A' and A coincide (of course coinciding then with a point of (c)), then the relation becomes:

|AB'|/|AB| = r/|OB|.

This is used in the discussion of the pedals of a point and its inverted on an Apollonian circle, in the file ApollonianPedalProperty.html .

Inverse Lengths

See Also