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 UV + VW = UW, where U, V, W are collinear with V between U and W.

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 A'B' + B'C' = A'C'. At this point we recollect the Angle Preservation Property of the inversion and its consequence - the distance formula:

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.

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 A'B' + B'C' = A'C' we obtain the triangle inequality A'B' + B'C' ≥ A'C'. The same derivation then tells us that in the more general case Ptolemy's equality becomes an inequality

AD·BC + AB·CD ≥ AC·BD.

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

Ptolemy's Theorem

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