# Ptolemy by Inversion

A wonder of wonders: the great Ptolemy's theorem is a consequence (helped by a 19^{th} 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 19^{th} 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' = r^{2}·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' = r^{2}·AB/(DA·DB),

B'C' = r^{2}·BC/(DB·DC),

A'C' = r^{2}·AC/(DA·DC),

which after a substitution and simplification give the required formula

AD·BC + AB·CD = AC·BD.

### Remark

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.)

### References

- 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)

### Ptolemy's Theorem

- Ptolemy's Theorem
- Sine, Cosine, and Ptolemy's Theorem
- Useful Identities Among Complex Numbers
- Ptolemy on Hinges
- Thébault's Problem III
- Van Schooten's and Pompeiu's Theorems
- Ptolemy by Inversion
- Brahmagupta-Mahavira Identities
- Casey's Theorem
- Three Points Casey's Theorem
- Ptolemy via Cross-Ratio
- Ptolemy Theorem - Proof Without Word
- Carnot's Theorem from Ptolemy's Theorem

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