has disappeared by dividing both sides of the equation by it. Now by using the sum formulae, and , it is trivial to show that both sides of the above equation are equal to Q.E.D. Here is another, perhaps more transparent, proof using rudimentary trigonometry. Define a new quadrilateral inscribed in the same circle, where are the same as in , and , lying on the same chord as , is defined by , . Then, has the same edges lengths, and consequently the same inscribed angles subtended by the corresponding edges, as , only in a different order. That is, , and , for, respectively, and . Also, and have the same area. Then, Q.E.D. Proof by inversion Choose an auxiliary circle of radius centered at D with respect to which the circumcircle of ABCD is inverted into a line (see figure). Then Then and can be expressed as , and respectively. Multiplying each term by and using yields Ptolemy's equality. Q.E.D. Note that if the quadrilateral is not cyclic then A', B' and C' form a triangle and hence A'B'+B'C' > A'C', giving us a very simple proof of Ptolemy's Inequality which is presented below. Proof using complex numbers Embed ABCD in by identifying as four distinct points . Define the cross-ratio . Then with equality if and only if . This proves Ptolemy's inequality generally, as it remains only to show that lie consecutively arranged on a circle (possibly of infinite radius, i.e. a line) in if and only if . From the polar form of a complex number , it follows with the last equality holding if and only if ABCD is cyclic, since a quadrilateral is cyclic if and only if opposite angles sum to . Q.E.D. Note that this proof is equivalently made by observing that the cyclicity of ABCD, i.e. the supplementarity and , is equivalent to the condition ; in particular there is a rotation of in which this is 0 (i.e. all three products are positive real numbers), and by which Ptolemy's theorem is then directly established from the simple algebraic identity Corollaries In the case of a circle of unit diameter the sides of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles and which they subtend. Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend. We