Ptolemy by Inversion
A wonder of wonders: the great Ptolemy's theorem is a consequence (helped by a 19th century invention) of a simple fact that
For the reference sake, Ptolemy's theorem reads
Let a convex quadrilateral ABCD be inscribed in a circle. Then the sum of the products of the two pairs of opposite sides equals the product of its two diagonals. In other words,
AD·BC + AB·CD = AC·BD
The 19th century tool we'll use is the inversion. We invert the whole configuration in the circle with center D and (some) radius r:
By that inversion, the circumcircle of ABCD maps onto a straight line with images A', B', C', of A, B, C, respectively. For those images we do have
P'Q' = r2·PQ/(OP·OQ)
that relates the lengths of segments PQ and P'Q' where P' and Q' are inversions of P and Q with center O and radius r.
Applying the distance formula to A'B' + B'C' = A'C' we obtain
A'B' = r2·AB/(DA·DB),
B'C' = r2·BC/(DB·DC),
A'C' = r2·AC/(DA·DC),
which after a substitution and simplification give the required formula
AD·BC + AB·CD = AC·BD.
Inversion with center D is useful even when the four points A, B, C, D are not concyclic. In that case, of course, points A', B', C' are no longer collinear so that instead of
AD·BC + AB·CD ≥ AC·BD.
(A dynamic illustration is available on a separate page.)
- H. Eves, A Survey of Geometry, Allyn and Bacon, Inc., 1972, p. 132
- Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994, pp. 155-156
- A Decade of the Berkeley Mathematical Circle, The American Experience, Volume I, Z. Stankova, Tom Rike (eds), AMS/MSRI, 2008, pp. 17-18
- S. E. Louridas, M. Th. Rassias, Problem-Solving and Selected Topics in Euclidean Geometry, Springer, 2013 (48-49)
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